Tom wrote, "I can't see any reason to do anything like rotate phase or
control
ratio, since the effects would be so small."

One reason to rotate the pattern (however one does it) is to use the
nulls to kill an interfering signal when receiving. Other than that,
there's certainly not much reason. Even with one-or-the-other of a
pair of crossed dipoles, at 45 degrees azimuth from max on a perfect
dipole pattern you're only down about 4dB from max, and if you add in
the feed-both-in-phase (or 180 shift), the patterns cross at about half
a dB down from max. Note though that the pattern for both fed in phase
is a bit broader than that of a single dipole.

Harry: you can simulate all this with the free demo version of EZNEC,
I'm sure.

Cheers,
Tom