I believe the term was introduced in 1985 upon election of Mikhail Gorbachev as the Secretary General of the Central Committee of CPSU (the position he continued to hold for more than six years). So, it's not really "post-Communist movement", it started by them in an attempt to modernize the regime. "Restructuring" is the most accepted translation, as far as I could judge.
When used in English texts, it always refers to that process of restructuring of the economy, administration, and often along with "glasnost' " ("openness"). I don't believe there is any particular reason to capitalize the word in English text.
In Russian language the word did, as the result of the events of the second half of 1980s and beginning of 1990s, gain some special meaning, which probably still lingers and will do as long as people who remember it. Eventually it's hopefully going to regain its general meaning. Compare with "The Prohibition" or "The Depression" in the US. Since Russian language lacks articles, capitalization is a way to make the word special, but only in writing.
Somebody living now in Russia should shed some light on possible negative connotation.