Updating speaker cabinets, how much does shape matter?

I have several sets of good speakers that I've rescued. Most needed nothing more than a refoam and recap, but some have truly awful cabinets.

In each case, I have everything, the drivers, crossover and the cabinet and stuffing to take measurements from. Some speakers (Allison, Epi) are rare-ish and unique, but unplayable due to enclosure damage. I'm somewhat handy at woodworking and design things for a living, so I want to try my hand at re-doing these speakers.

My question is this; I want to create new cabinets but don't necessarily want to keep with the current box's shape. If I keep my internal volume, how much does shape matter in a speaker enclosure?

How much of the old box designs take advantage of the baffle design? For example, the Allison Six is one cubic foot in size. The tweeter is positioned on one 12x12" face. Does the tweeter use the face to reflect the sound, or can it be mounted in other configurations?

I hope my question sounds clear, it's still rather unformed.
Thanks in advance!
There are several ways in which the enclosure affects the sound. I will list what comes to mind first.

1) Internal Volume - affects how deep a woofer can play. You have some headroom with a sealed enclosure but the correct volume is critical for proper vented design. You can run some simulations with WinISD to get an idea of how the internal volume affects the sound.
2) Baffle step - baffle size and distance of driver relative to the baffle edge will affect the sound. Some crossovers account for that, some don't. The Edge software can run simulations for baffle size and placement.
3) Tweeter flush mount - sharp irregularities on the baffle will affect the high end.
4) Cabinet Resonances - Bass cabinets need to be cross-braced so they don't vibrate.
5) Time Align - Speakers are time aligned when the voice coils of all drivers align on the same plane. Almost no crossovers ever compensate for that but it's worth to mention.

It's senseless to redesign the cabinet if the crossover already addresses any of these issues. You need to find out what exactly the crossover does and then you will know what parts of the box you can modify.