Spinning off from this more specific question:
Russian visa bureaucracy is infamous, and one of the quirkier bits is the requirement to have a letter of invitation (LOI) for virtually all visa types, including tourism visas, although then it's called a "tourist confirmation" instead.
However, as far as I can tell, if you have APEC Business Travel Card, you're eligible to enter Russia without a separate visa or a letter of invitation -- as long as you're visiting for business. If this is the case, how can you prove that you're visiting for business? Or are you best off bringing an LOI anyway?