The only time I would use a 6" sub would be for a computer desk setup. Otherwise, a 6" just wouldnt have nearly enough output for any medium+ size room for me. To be honest, a 10" sub is the minimum size sub I would ever recommend in any sort of home theater.
Of course it also depends on the quality between the 2 subs. If the 6" sub was a really well made sub & the 10" a junker then that's a different story. I have no experience between the 2 you are asking about so the first paragraph I typed is subjective.
The 6" specs may say it goes lower in comparison to the specs of the 10" but you can't compare the 2 subs that way because we don't know how loud those frequencies are being played. Even if we had the -3dB (F3) it still wouldn't tell us everything. So I would do this: play a few action scenes from some movies & play them at the loudest you would typically listen to them at. Make sure each of them are adjusted properly & listen to which sounds better to you. The 6" being smaller probably wont get as loud or as low as the 10". However it's not always about which is the lowest/loudest sub. Listen for which SOUNDS better to you!
There are a lot of bass heads around here & for a lot of us a 10" is too small for us. But you need to choose the sub that sounds best to you!
I see in your sig that you have both subs. Do you own both & are trying to decide which to use? Or are you trying to figure out which one to buy?
Right off the bat my thought is " Use the 10"!" but like I said, there is more to it than just driver size....
Hope this helps ya!