Wow! Ask and ye shall receive... Thanks! It works here with the inspector minimized, it pops open if it wasn't already. The only way you have to do it twice is if the inspector is not open and set to ProChannel, in which case you're toggling it off and then on again.
This is nowhere in the keyboard shortcuts dialog, at least not that I can find. Rather, if you happen to know it already and search by Locate Key, it's there, but if you search by function, it doesn't show up. This would appear to mean that it can't be bound to a different key combination if one wanted to customize it - it's hardwired to this combination. Not sure why that should be, but I guess I can live with it.
Hmm... Are there also unlisted bindings for clip properties and track properties? Why yes, there are: Shift-I and Ctl-Shift-I, respectively. Oddly, they behave slightly differently in that if the inspector is closed with either of them selected, they will toggle the properties dialog off while opening the inspector. Strange, and I'm not sure which behaviour I prefer, but I guess it would be better if they were consistent...