The Cuban missile crisis is less a case of one side backing down than you may be thinking. The Soviets demanded and received a public declaration that the United States would not invade Cuba if the missiles were removed. The US also made private promises to remove missles from Turkey in exchange, and followed through. An American spy plane was also shot down while the crisis was ongoing; the US did not acknowledge the loss.
https://history.state.gov/milestones/1961-1968/cuban-missile-crisis
When I was a sociology undergrad, the Cuban Missle Crisis was a frequent subject of study, as an example of effective conflict resolution. An example of the importance of dialogue, of concessions on both sides, and of allowing both sides to save face to resolve conflicts effectively.
History is certainly in the eye of the beholder in this case. You mention that you were taught that the US stood up to the Soviets and they backed down. This is the way in which most US textbooks depict events such as the Cuban Missile Crisis, The Berlin Airlift, and Afghanistan. However, as in much of history, textbooks do not have the space, or are not allowed the space, to truly go into the nuance of understanding the past. First, let's look at some of these major Cold War events in which the Soviets "backed down." Then, we will look at other events and how they are depicted. I study textbooks, so I will be focusing on how textbooks in US high school's portray the events. I will be using Prentice Hall's World History. Primarily because I have it on hand.
Context
Following WWII, the breakdown of relations between the USSR and the US was unexpected, at least the level of animosity was relatively unexpected. Hence, Berlin was a particularly odd thorn in the side of the Soviets. As the Western occupiers began to fuse their zones, the Soviets refused to join in since Stalin saw a unified Germany as a threat against the USSR (the Soviets believed that the US wanted to use Germany against the USSR).^1
Berlin Airlift
This is pretty much the most clear example of the Soviet Union actually failing to accomplish its goal. World History depicts it as an attempt to remove the Western Allies from West Berlin. It doesn't mention that the immediate cause was the strong currency in the West and how it endangered the East due to West Berliner's using the currency while East German's had a weak currency. They could see the difference between the currencies due to nature of the occupation in Berlin. Ultimately, after more than a year of the airlift, the USSR backed down on using the blockade, but not in its goal of gaining West Berlin.
Cuban Missile Crisis
Prentice Hall depicts the end of the Cuban Missile Crisis in this way, "Soviet Premier Nikita Khrushchev agreed to remove the Soviet missiles, and war was averted." Although "backed down" is never used, it is clear that the text depicts Khrushchev as gaining nothing for removing the missiles. As others have notes, however, the US agreed to remove missiles from Turkey. Hence, the narrative is certainly accurate, but missing key information that would make the incident one of negotiation rather than just brinksmanship.
Afghanistan
Afghanistan is an area which the books don't actually depict as the USSR "backing down." Instead, they usually relate it to Vietnam. They note that in both cases, traditional military forces were worn down by guerrilla fighters and the morale at home was exhausted. One must also remember that Afghanistan was not an example of the US and USSR facing one another down as were the previous two examples. It was a war in which the USSR's military was involved. A loss in a war does not equate to "backing down."
US Context
One must remember some of the guarantees given by Stalin and the Soviet Government during the agreements made during WWII. During these conferences, free elections had been promised by Stalin. Thus, a goal of the US was the liberation of ALL of Europe. With this in mind, lets look at a few events.
Czechoslovakian Coup
Following the liberation of Czechoslovakia, the government-in-exile, led by Edward Benes, maintained democratic control of Czechoslovakia. However, socialist and communist parties gained a great number of votes and Benes made a coalition with them. Due to the US removing a loan from the Czechs, the leftist parties made more gains. Soon, other parties were pushed out of the coalition and Benes was forced to resign (his former foreign minister MAY have been assassinated). The communists took control in a bloodless, but illegitimate takeover (rigged elections followed the takeover).
In response, the US & Britain gave lip service to the Benes government, but did not take any actions to support democracy in Czechoslovakia. This is just one example of the ways in which free elections were ignored or fixed and communist, one-party governments installed in Eastern Europe. In each case (Poland, Hungary, Czechoslovakia, Romania, & Bulgaria), the US did not act and "backed down" on their demand for free elections in Europe.
The Berlin Wall & the Berlin Crisis (Showdown at Checkpoint Charlie)
In 1961, a series of events of events can be depicted as the US having "backed down" when the USSR made demands. The first is in regard to the status of Berlin. The city itself was protected as a an occupied zone through the Four Power Agreements. This gave the Western powers access to West Berlin. Furthermore, the Western Powers had certain privileges of access throughout the city, including the Eastern zone. Finally, due to its control by all four powers, Berlin was a location through which up to 4.5 million East Germans had escaped. The Soviet Union and East German government, therefore, saw West Berlin as vital to the security of East Germany. Both preferred to gain control of all Berlin, but sealing the escape route was also essential.
Hence, in 1961, when President John F. Kennedy met with Nikita Khrushchev for the Vienna Summit, the table was set for confrontation. Khrushchev had previously threatened to sign a treaty with East Germany giving that state control of access to Berlin. They could then shut out the Western Allies and force a confrontation or give them access only through agreements with East Germany. It would also be a violation of the Four Powers Agreements that did not give each power the rights to stop occupation on its own. The meeting did not come to a firm conclusion, but important was that Kennedy took a stance no previous president had ever taken. Kennedy, by his language during the conference, forfeited US "protection" of East Berlin that was inherent in the division of occupation. He continually used the term "West Berlin" when speaking of US interests and determination. This signaled to Khrushchev that changes in East Berlin and East Germany would be allowed by the US as long as West Germany and West Berlin's status were not altered.
Hence, the Vienna Summit has come to be seen as a location in which JFK gave a concession to Khrushchev from the very beginning. This concession likely led to Khrushchev's later actions in Berlin and Cuba.
In response to this signal given during the Vienna Summit, Khrushchev gave the go ahead for East Germany leader Walter Ulbricht to construct a wall surrounding West Berlin. This began on August 13, 1961. The US response to this was limited to criticism, but there was very little action. Perhaps the only thing done was that General Lucius Clay, who had been commander in Berlin during the Airlift, was made the senior American official in Berlin. This act did suggest a stiffening of US resolve to some degree.
Hence, by late 1961, the US had backed down regarding the rights of East Berliners and had left them on their own. Nevertheless, World History only stated, "The wall showed that workers, far from enjoying a communist paradise, had to be forcibly kept from fleeing. Hence, even an incident where the US gave ground to the USSR was depicted as a victory for the US.
Sources:
Kempe, Frederick. Berlin 1961: Kennedy, Khrushchev, and the Most Dangerous Place on Earth. New York: Putnam Press, 2011.
Ellis, Elisabeth and Esler, Anthony. World History. Boston: Prentice Hall, 2007.
Carmichael, Neal. "A Brief History of the Berlin Crisis of 1961" National Archives. National Declassification Center.
I wouldn't characterize Cuba as being a unilateral backdown (as the another comment indicates, there was mutual compromise involved), and I wouldn't consider Afghanistan a "backdown" per se, unless you also want to call Vietnam a "backdown."
The trick about the "backdown" approach is that it is not entirely clear what one means by it. Was the Korean War a "backdown"? If so, by who? One man's "backdown" is another man's "cutting losses" or "declining to escalate." The framing of it as "standing up" versus "backing down" seems to privilege aggression as the positive approach.
There are several incidents where the US definitely declined to escalate, though, if that is what one means. The Bay of Pigs invasion is one very famous one. The Suez Crisis is another.
So maybe the Cuban Missile Crisis isn't the best example of the Soviets "backing down," as some other commentators have noted. I'd agree, but I think it's worth pointing out that the missiles in Turkey that JKF agreed to withdraw were obsolete the moment they were deployed (and therefore were likely to be removed anyway) and that the Cuban Missile Crisis, rightly or wrongly, was an embarrassment for Khrushchev which played a role in is overthrow just a few years later.
There is, however, at least one prominent example of the United States "backing down" to the Soviet Union after WWII (and obviously examples of the United States "backing down" during the war to Soviet demands regarding what would become the "Soviet sphere"): the complete abandonment of "Rollback," a policy promoted at some point by both Truman and Eisenhower aimed at liberating nations from Soviet control. The policy manifested in the Korean War. And while the contrasting policy of "containment" (i.e., the suffocation of Soviet rule by preventing its spread) had gained prominence in American foreign policy by 1956, the National Security Council nevertheless responded to protests in Hungary against Soviet rule by reaffirming that "the elimination of Soviet domination of the satellites is . . . in the fundamental interest of the United States." Indeed, the United States had attempted to spur such unrest in Hungary for years, dropping propaganda in Hungary asserting that Soviet rule could be overthrown.
Once revolution broke out in Hungary, however, the United States quickly backed down. Sure, the Eisenhower administration initially protested the ensuing Soviet crackdown and promised not to ally with Hungary if the Soviet Union allowed the nation independence, but this was the extent of the American effort. The United States declared that it would not get involved in the military conflict, which, according National Security Adviser Walter Rostow a decade later, gave the Soviets a free hand to crush the rebellion.
Now you might expect to see lots of American historians counter this argument by saying that rollback was never official American policy, that the concurrent Suez Crisis is what really stopped the United States from doing more, etc., etc., but you should see how Hungarian historians remember this apparent change of heart. As Laszlo asserts, "The American response to the 1956 Hungarian revolution encapsulated Washington’s Janus-faced attitude toward the liberation of Eastern Europe." In any case, there is no doubt that the United States had backed down from its bold policy of "rollback" in favor of policies that would not lead to a direct confrontation between the United States and the Soviet Union.
The article I cite: http://www.coldwar.hu/html/en/publications/rollback.html