Audible "click" on split. Huh?

Solution 

Curiously, additionally gluing the item BEFORE step 2 seems to prevent the click. Though I'm unclear why.

Doesn't happen here. A couple of thing you could check: 1. Preferences -> Project -> Media Item Defaults check that "Overlap and crossfade items when splitting" is off. 2. Right-click the toolbar button for grid and tick "Snap to project sample rate". Other than that I'm not sure what could be causing the click.

Yeah, this sounds like a crossfade setting,

If you render the affected item(s) is there a one sample clip at the start of the crossfade when you zoom in all the way? Maybe try the js dc offset plugin on the affected track or item(s). Does the click disappear? Out of curiosity, does this click knockout the RMS meter on the master bus?

When I split, I split at zero crossing... no more clicks.

Noooo, these didn't solve the problem. I can have a loop that works perfectly when I test it, but then produces a click when I glue the self-same items into a new file.

I had some clicks happen when I had extraneous time signature markers and had time-stretched my items in various ways. When I glued them, funny stuff sometimes happened. Basically I think I had too many things messing up my samples, too many splits and stretches going on. It was no wonder. Redid it in a more elegant way (after removing time signature markers) and it was OK.

Could this be a hard drive or computer performance problem? When you glue items a new wav file is created which can be easier to read than a combination of audio clips all pulling from different source audio. Perhaps try increasing the buffer size?