Most people intuitively think LFE is an independent channel. It is not technically. The best case to explain this is the ending credit sound tracks of Fast and Furious 2. It is so good that I bought a CD sound track and completely disappointed. It simply does not have enough amount of clean bass energy recorded on the CD. It completely makes sense from dynamic range point of view. In CD, there is only 16bit resolution. If that entire resolution is occupied by the slow moving bass signal, there wouldn't be room left for other music signals. Since hearing sensitivity is very low at low bass, that can take away significant dynamic range. So LFE comes into play. Signals below a particular frequency (for instance 120hz) are filtered and put in a single LFE channel. During playback, LFE is actually boosted by 12db (or full 2 bits). In other words, one can imagine the original sound recording is recorded in at least 18bits. Filter out say 120hz and below and the remaining should not exceed 16bit resolution and can be recorded in 16bits for front left and right. The LFE is then reduce by 12db (or reduced by 2 bits) so that it can also be recorded in 16bits. The advantage is when the signal is reconstructed, we want to get the 18bit resolution back with a clear bass reproduction. But literally it is same principle as xover design. it shows the LFE channel should be added back to their original channels. Unfortunately, it is impossible to encode which signal is from which channel in LFE. So most 5.1 decoding method add LFE to both front left and right channels. Then do the filtering again for subwoofer channel output. In order to avoid extra signal leaking from left channel to right and vice versa after decoding, the movie company should keep their "encoding" xover point for LFE as low as possible, such as 80hz. On the other hand the decoding LFE filtering does not make sense at all. The only possible explanation is as a way to prevent digital noise (because LFE has a very low sampling rate) or other artifacts from entering front channels. In short, one should think of 5.1 or 7.1 as an encoding/decoding method, not as formats directly mapped to 6 or 8 physical channels.
Even though the above is based on music signals, it applies to movies too. Special effects such as explosion and gun shot have high frequency components that need to go to front and rear channels. So put LFE back to their original channel to get a full spectrum special effect is very important. As a result, AVR should first add the LFE signal back to front channels and then do a xover (or bass management) to provide one single SUB output.