This car runs like a top (ordinarily), has 113k miles on it and a
Fairly new battery. It gets regular maintenance and was last used
Yesterday.

We tried to start it this afternoon, but only got a whirring sound.
Not the usual cranking, or weak cranking or the clicking if there is an
Overt problem - just "whirring."

All the panel lights go on.

Where do I look? I suspect the alternator but have done no substantial
Car work for 20 years, so I have no useful electronic test devices, only wrenches, etc.

Your description sounds like the over riding clutch in the starter motor is bad. You will likely have to replace the starter motor.

No, I'm afraid that's the sound of lost compression. Your timing belt has stripped some teeth off it and now gone out of time with the cam.

Thanks guys.

The system won't let me leave either of you feedback "until I spread more feedback around."

Well guys, it ended up that it was the battery. Eventually, I got no response at all to turning on the ignition, not even the lights on the displays. Jumping the battery from the one in my running Avalon did absolutely nothing. I trickle charged my spare type 35 battery and as a last resort before calling for a tow, jumped the Camry from the spare battery. It started, much to my surprise, since I thought I had made good contact when I was jumping from the Avalon. Ran the car over to my mechanic, who was too backed up to look at it for a few days. He eventually told me it was just my battery, after testing the starter, as it was down to delivering just 10 volts and losing charge rapidly. So, thanks for the suggestions. I'm hoping it is not an intermittent starter and it appears to definitely not be the timing belt. We installed the spare battery and I was good to go.

Bravo! Sounds like it was cheaper than my idea.

So far - about 6 days - it's been running like a champ. Bob (my mechanic) does only toyotas and takes no new customers. Charged me all of $25 for all the time he spent and to install the battery.