OK. Heard back and resolved the issue, it seems.
Not at all what I'd expected.
Windows 7 x64 at church and home, with a Samsung Android tablet. IE 8, with Opera as the browser of choice.
Removing AdSense resolved the issue. Squared it away on my phone, too - no need to select "Desktop view" now.
I found a few things in searches about AdSense causing blank screens. I'll keep looking further for why this would be an issue. I'm especially wondering about the ad-randomizing function in relation to AdSense. Maybe those conflict. It could be that leaving AdSense but removing the randomizing function would work, too. All sorts of combos to try, and since the user's system is Win 7 x64, all I'd have to do is reset to IE 8 to see if I see the same.