Thanks for that reply. You're asking a lot of good questions.

The computer, according to the mechanic, doesn't show anything abnormal at all. If it did, I suppose we'd be in better shape.

The charcoal canister idea is intriguing (to tell you how little I know about cars, I don't know what a charcoal cannister is, or what it's for), and I'm going to take it up with the mechanic. When it's towed, it's generally on a flatbed, though the angle does get shifted around some as it gets loaded and unloaded from that.

The starting failure has occurred both with the car parked flat and at an angle. It gets parked in all sorts of different places (we live in a big city where parking is catch as catch can) and has failed being parked uphill and flat. I don't recall it failing while parked downhill, but I don't park downhill as often, so I wouldn't conclude much from that.

The mechanic uses the same key I use. I don't think it's a difference in how each of us starts the car; when I start the car at the mechanic's garage, it starts just fine.

When/if I figure it out, I'll definitely post it.