Everything I've ever read on this subject sez it's even OK to mix them any which way you want, avoiding extremes as, say, mixing a 0-15 with a 20-50 for any extended periods of time or above average loads. The purpose of oil is to prevent metal-to-metal contact and even a 39 cent Pep Boys SAE 30 special will do that; in a pinch, that will get a rice rocket home, but I wouldn't take it to redline...
Switching back and forth is A-OK, but why would you even want to use anything but synth? Well, unless you got a Harley...
Your Jedi mind tricks won't work on me, Skywalker.
2004 Kawasaki ZR1000A2 Z1000
1996 Suzuki GSF-600S Bandit
1995 Kawasaki ZG-1000 Concours
2002 Shinto Burakumin