Thanks--I'll keep all your tips in mind. Btw, I don't know about others, but for myself, I often fail to pay attention to the oncoming left-turning cars (the same when I am a pedestrian crossing a street), as my attention is devoted to the cars behind or passing me.
You have to know who's ina position of threat which changes as you approach an intersection. Early as you're first approaching, you need to worry about a passing car right hooking you. However, at a certain point that isn't a problem since they can't get around you, and the chances of one turning right into you are very low. So you have to give attention to the possible left cross, especially in NYC where drivers sometime try to make a left shooting a gap in traffic.
Also, do a check to make sure crossing traffic with a red is actually stopping, until they confirm that by slowing, you have no idea what they'll do.
All of this, checking shoulder, left front, and both cross directions is a quick head swivel, and there's plenty of time at bicycle speeds. You just train yourself to run the checks the way a quarterback checks of his receivers working down to backup targets.