Subwoofer hook up

I have a pretty basic question but I keep getting different answers from people. I have an AVR for movies and an integrated amp with a home theater bypass for when I listen to music. I am looking to replace my REL Storm III with a new subwoofer. The Storm uses a "Neutrik" connector which may be proprietary to REL. It connects to the subwoofer directly from the speaker outputs on the amp. A new subwoofer will not use this connector and I have the option of connecting it via the AVR or via the pre-outs of the amp. Which is correct. I want to listen to music without turning on the AVR (but obviously using the sub) and I need the AVR when watching movies. I really don't want multiple cables and switching devices as some have also suggested.

You can do that with the REL Storm which has independent High Level (neutrik speaker connection from amp) and Low level (RCA from AVR). However, other subwoffer manufacturers usually do not offer independent High and Low level inputs, so without knowing the sub you are replacing the REL with, it is hard to give an answer/advise.
You might try downloading manuals from the web sites of subs you're interested in.They will have diagrams for hook ups in various systems.
This can be done - I have done it, but it depends on the hook up options on your equipment.

1)your sub has separate inputs for LFE *and* L/R Line Level inputs (not a combined LFE on Left input option)

2)your integrated amp with HT Bypass has line level pre-outs.

If so, all you need are some interconnects to go from your AVR Sub out to the LFE input on the sub and a pair of ICs to go from the pre-outs on your int amp to the L/R line level inputs on the sub.

For Home Theater, the integrated amp gets switched to "HT Bypass" and the pre-outs should be disabled as the preamp section of the int amp are bypassed and it acts as a power amp only for your Front L/R speakers.

The sub LFE input will get the bass managed signal from the sub out on the AVR.

Enjoy the movie.

For 2-channel music listening, turn off the AVR - there will be no sub out signal going to the sub.

Select whatever source you want to listen to you on your int amp and the preouts will feed the sub the L/R signals. Of course, you will have to manually integrate the sub xover with your mains and your mains won't be high pass limited and will get the full range signal from the int amp - but I have found that with time and effort you can properly integrate mains and sub this way esp for music, since that won't have any scary and dangerous LFE signals ;-)

Enjoy the music.

No switches, no cable swapping - slick as could be.