Single Sub Woofer for two separate systems?

You can do what you want if you have or can find an NHT X-2 XOVER.

You can.

I do - works perfect.

AVR sub out goes to the LFE input on the sub.

Integrated amp with HT Bypass with preouts goes to the line level inputs to the sub.

For movies, Int Amp set to HT input, preouts are disabled, and the AVR sub out drives the sub with the scary low LFE effects.

For music, AVR is off, and source components are directly connected to the int amp, sub gets full range audio and the xover is set to blend with my mains.

No feedback, no cable swapping.

Everything Erikt said, absolutely! EXCEPT you can't do what Erik says UNLESS your sub is equipped with L and R line level (2-channel) inputs and a separate (single) LFE (low frequency effects) input. I know the MartinLogan Depth and Descent are configured this way, and probably many others these days -- but check first!
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You can do what you want. Just confirm that your sub will allow you to use 2 seperate systems. I have a pair of Velodyne DD10Plus that say that you can do that.

How about a Niles AXP-1 input selector? Take the subwoofer outputs from each system and plug them into an input on the Niles AXP-1. Then from the Niles AXP-1's output plugs connect that to your subwoofer. When you want HT or your other set-up just change inputs on the Niles. I use the Niles now and don't have any sound degradation issues. It is built quite well too for the $ IMO. You may or may not need some RCA F to 2 RCA Male Y adapters too. I had some laying around so I used them with my Niles. Good luck.