The Soviet Bloc countries were generally off limits to Western operators, with some exceptions:

• Berlin Corridor, which crossed East German airspace brought aircraft through airspace under Soviet control. During the Cold War, this corridor was open only to the flag carriers of Allied-occupied Germany (US, France, UK). This was an interesting aspect of the Cold War because Allied air traffic controllers in the Berlin Air Safety Center worked alongside Soviet military representatives to pass information back and forth.

• Corridor from Frankfurt/Copenhagen/Helsinki to Moscow and Leningrad. This was primarily a Baltic route. Finnair was the first to be allowed into the USSR (1956) and Pan Am and Air Canada were allowed later in the mid-1960s to serve Moscow.

• Yugoslavia, being one of the most progressive of the communist nations, allowed Western aircraft to transit from Austria to Greece and back with few restrictions.

• Trans-Siberian service (described in the article below) was implemented in 1967 with a codeshare Tu-114 (and possibly DC-8) flight with JAL (Japan) and Aeroflot. This allowed flights from Japan across Khabarovsk and deep Siberia to Moscow. Other Western operators were allowed use of this route as long as a technical stop was made in Moscow. All other airlines during the Cold War era flying via Europe and east Asia either had to either go the polar route via Alaska (Korean Air, JAL, etc) or via the Silk Road route through southwest Asian countries (British Airways, Air France, etc).

• Air India was allowed to operate a southern route to Moscow in the 1970s via Tashkent. This route exited the USSR in Afghanistan. I'm not sure if other Western airlines were able to use this but they would have been bound to stop at Moscow.

All crossings over the USSR required a Soviet crewmember to serve as navigator and often to speak with ATC on the ground to assure positive control of the aircraft. As Russian was an official ICAO language, the USSR had no mandate to provide English ATC services, and English speakers were rare in some of the remote Soviet flight information regions. A Russian translator was required even as late as the mid-2000s on some of the new north-south Siberian routes, and I recall Jeppesen shaded the regions on their charts which were off-limits without a translator.

SOURCES:
An outstanding one is Aviation Week in the 4/6/1970 issue and 4/13/1970 issue. Requires login but you can access it with a free registration.