Maybe I'm taking you too literally, but I wouldn't want to have a handoff sync-up (or any meeting, really) on a Friday night, nor push that earlier so significant things can happen between sync-up and the actual shift in responsibility from person to person. Friday-to-Friday does sound good.

One thing I really liked in a previous job was a split daytime-vs-nighttime rotation. It was well worth a little annoyance to set up in our tools. One week you'd be the 'daytime' oncall for business hours (something like 9-5 Mon-Fri, though we might have tweaked those hours a bit; it might have been 10-6 or something). The next you'd be on call for the complementary time (5-9, weekends). You were on call for the same total amount of time, just smeared over two different weeks. It ended up being less of a burden to optimize your schedule for a reasonable response time, but operational work still got done. And in practice awareness of operational issues was not too hard to maintain between the two members of the split.

(I think the best thing, if you can swing it, is probably a follow-the-sun rotation where there are three teams distributed 8 hours apart around the globe, and they trade off 8-hour workday shifts. But a lot of uncommon things probably have to be true of your organization for that idea to even be on the radar.)

We split in-hours and out-of-hours, and I wouldn't want it any other way. It's especially good if a late night incident keeps you from sleep, because you can take the time back the next morning and let someone else pick up the pieces :).