My understanding was that the watt-hour meter actually measure VA-hours.
I
asked the electric company once, and that said that was the case. But I
wasn't speaking with an engineer.

That's VARs, not VA-hours. For 'volt-ampere reactive'.

If you can have watt-hours, you can have VA-hours. Both are units of energy.

Is VAR a unit of energy? No, it's a unit of (reactive) power. Energy and
power are not the same.