Did the Soviets have anything similar to the U2 or SR-71? And did they fly over the US in the way the US did over the USSR.

The question was asked and answered here.

Yes and no. They did have a U2 like plane in the M-55. However Russia had issues spying on the US via planes. Problems the US did not have. That issue was the Atlantic and Pacific Ocean. You have to understand the the Russia is a huge country that UNLIKE the US is bordered by a vast amount of land not all of which is friendly. The US is ocean locked on the east and the west while they have an ally and interconnected military with their northern neighbor Canada. The US was able to send spy planes not just over Russia but would send planes that flew on the BORDER of Russia. They could do this because UNLIKE Russia the US had basses all along their border. In Turkey, Germany, Japan, and throughout the pacific. This made spying runs easy and more importantly not as fuel costly since you just took off and in no time bam your on the border ready to snoop. So spy planes where able to spy on the Russia with ease not ever entering foreign airspace. On the other hand Russia would have to fly across the pacific, past all the detection rings in the pacific and somehow gets some pics before the jets arrived. The only other way was to go through Alaska and Canada which again good luck detection posts in Alaska as well as joint military bases in Canada will get ya. Russia focused on Live intelligence gathering due to the openness of US culture while the US focused on physical intelligence due to its ease and the difficulty of penetrating a closed nation like the USSR.