Yes! In fact one of the heros of the Tsarist Army during the war, General Aleksei Brusilov, joined the Red Army in 1920. However he was not the only one to do so. Many were influenced to join with the Reds because of their Russian patriotism. Their sense of duty to the Russian nation preceded their sense of duty to the aristocracy and tsarism. Brusilov personally felt that the period of Bolshevism would not last very long, but even so for now Russia had chosen communism, and it was more important to follow Russia, than to try to fight it.

A key turn for those nationalists who joined the Reds was the invasion of Ukraine by Poland. They did not want to see Ukraine, a territory they felt to be innately part of the Russian nation, fall into the hands of the Poles. With the Poles invading on one side, and the White Army fighting on the other, it seemed that the Whites were putting themselves before Russia. The Reds were able to use this to their advantage, and began to use somewhat patriotic propaganda to bolster support for the war effort against Poland, a sort of early version of what Stalin would later use during World War II. This also spawned the ideas of "National Bolshevism," whose supporters thought they would be able to turn the Soviets into true Russian patriots from the inside.

Brusilov personally used his influence within the new regime to try to save officers from execution, though was not always successful. After the war he was assigned to various positions overseeing the Red Army's cavalry. He was allowed to retire in 1924, and then spent some time in Czechoslovakia. However not wanting to be like the emigres who he hated (and hated him), he returned to the Soviet Union and died there in 1926, where he was given a hero's funeral.

Source:

Figes, Orlando A People's Tragedy: The Russian Revolution 1891-1924 New York: Penguin Books, 1996.