In 1971, the USSR (and satellites) voted in favor of UN Resolution 2758, which recognized the People's Republic of China (PRC) as "the only legitimate representative of China to the United Nations" (voting).
Why did the USSR do so, given the hostilities between them and the PRC?
I think this one could be answered on two levels, one a more general one which I'll attempt, and the other a very specific one that looks into the motivations of the Soviet leaders of the time. I'm sure someone has some notes by Khrushchev why this was a desired move by the USSR.
Some terminology should be clarified first. The PRC did not "enter" the UN; instead, the UN motion changed the "one China" whose credentials were recognized. "China" had been the ROC up until 1971 in international terms; in 1971, with this motion, "China" started to apply to the PRC instead. No new members were admitted or old ones expelled.
This motion to recognize the PRC instead of the ROC was, in essence, an anti-American statement due to the support that the US had been giving the ROC since the mid-1940's. Yet, there was a greater shift of alliances as well and though the US was one of the minority to vote for retaining the ROC, this time is described:
However, in the beginning of the 1970s, the United States saw the geopolitical opportunity to move closer to China in a strategic move against their by then common adversary, the Soviet Union. The United States eventually broke formal relations with the ROC only in 1979, but the strategic shift in the early 1970s, combined with a large number of newly-independent former colonies that had some ideological solidarity with Beijing, turned the tide once and for all against Taipei. Still, it was a combination of Taipei and Beijing’s longstanding opposition to proposals for both PRC and ROC representation in the UN, together with the global strategic changes, that led to the end of the ROC representation in the UN, and in consequence also to the ROC’s expulsion from all other major international organizations.
In more general terms, this was about a decade old controversy:
From the 1960s onwards, nations friendly to the PRC, led by the People's Republic of Albania under Enver Hoxha, moved an annual resolution in the General Assembly to expel the "representatives of Chiang Kai-shek" (an implicit reference to the ROC) and permit the PRC to represent China at the UN. Every year the United States was able to assemble enough votes to block this resolution. Both sides rejected compromise proposals to allow both states to participate in the UN, based on the One-China policy.
The admission of newly independent developing nations in the 1960s gradually turned the General Assembly from being Western-dominated to being dominated by countries sympathetic to the PRC. Not only the newly founded developing countries, but also most of the Western countries eventually decided to recognise the PRC. During the 1950s and 1960s, United Kingdom, Switzerland, Sweden, and France shifted their recognition of China from the ROC to the PRC. In the early 1970s, Canada, Turkey, and more western countries established diplomatic relations with the PRC, and severed diplomatic relations with the ROC.
In a Security Council meeting on 9 February 1971, Somalia objected to the credentials of the representative of Republic of China as China representation, and ROC and the United States responded that the question of China's representation should not be dealt with in the Security Council.
And those discussions led straight to the General Assembly Resolution 2758. Looking at a map of the vote, the opposition to the motion (red) was organized by the US while Europe supported the PRC—but also the USSR and all of the Eastern Bloc (note Albania, Poland, Yugoslavia, Mongolia, etc) voted in favour:
Lastly, perhaps it's relevant that while the PRC wouldn't necessarily vote in line with the USSR in the Security Council, the ROC's single use of their veto was against the Soviet measure to admit Mongolia into the UN.
