I searched a lot for an example of how to do pitch shifting with FFT in PD with minimal luck. I did finally run across a patch that said it could do what I needed but it was broken. I have fixed the patch (and its sub-patch) so that they now work correctly and have attached them.

While these do work, they don't do everything I want -- I would like to be able to shift frequencies by bands: 3000 to 12000 by a factor of 2 (600 cents?) and then repeat, for instance. If my patches are of use, perhaps someone else can contribute the banding??

http://www.pdpatchrepo.info/hurleur/fft-pitchshifter.zip

the lrshift object can shift samples in a block left and right, and can therefore shift fft bins up & down
also see http://www.katjaas.nl/pitchshift/pitchshift.html

Another option is tof/sample-shifft~.pd (sic!). It has both frequency and time expanding. I think I see how to add banding to either, but have not actually sat down to do it.... Lrshift won't do it alone but might work with [+!]. For instance, I think I could shift and add twice to get both the 2 and 4X divisions using two shifts (for instance, shift 1024 out of 2048, add, shift another 512, and add again.....] It would require making a copy of the array but only one. That would not be a general banding solution, but would satisfy my need!

For audio, I think I could ignore the imaginary if cpu load becomes an issue.

you shouldn't just ignore the imaginary part, at the very least get the magnitude of the signal and phase bash it (see the phase bashing example). you should know though that all the I07 example does is correct the phases of it's output so that the phase change between 2 output windows is the same as two input windows the same distance apart. The FFT part of the patch does not do any pitch shifting or time scaling (except for the phase shifting), that is done by reading from the buffer using granular synthesis in the example.
I can't really wrap my head around what you mean by adding and shifting (I'm not sure what you mean by banding actually), but you're right that lrshift~ won't do it alone, it's just a frequency shift. I think using a separate table is the way to go if you want to pitch shift using fft, e.g. if you want to pitch shift 2X take every bin and put it's magnitude value in the bin 2X up from that. Alternatively you can just 0 every other bin (the odd ones) if you want to preserve the spectral envelope.
In taking the bin number in order to pitch shift it, you can consider the phase from the phase vocoder in order to get the "real" frequency, meaning the frequency deviation from the bin. I'm not exactly clear on getting the correct bin numbers back after pitch shifting though
non-integer pitch-shifting is harder, I believe you need to interpolate
btw I can't really tell the difference between lrshift~ and spec2_shift~, do you know if there is any?
spec2_stretch~ seems to be able to do what you want.
also check out pvoc~ in the fftease library

A "band" is a frequency range -- in this case, I want to take the "band" between, say, 3kHz and 6k and divide by 2, similarly, the band from 6k to 12k would be divided by 4 (hence the two left shifts). I figure that if I shift the entire spectrum from 3K up left by a factor of 2 and then do the same thing I will have approximately what I want (and could repeat until there are no more frequencies that are non-zero).

I will do the shifting and adding to both real and imaginary -- I suspect the "cost" of computing the magnitude is at least as high as operating on both arrays.

Thanks for the pointer to pvoc~ -- I may have missed it in searching for examples of what I want to do.

I hope to have this running on a Raspberry Pi for a portable solution "real soon now."

I am trying to do the shifting to overlay the high frequency spectrum over the low frequencies. I tried lrshift~ but it seems to not be happy with using a variable for the shift amount. Spec2_shift does take a variable, but seems to not like negative shifts? In any case it fails for me even using the help.pd for it if I take the shift negative. I am not very good at C programming (Java is my language) and I could not figure out what the error is in looking at the code.

I can post my patch for testing, but the one that comes with spec2_shift~ is good enough.