Hang by tail , split part way up and around vent and tail skin , flesh and split after its fleshed . Can’t get the clean skinning right because I don’t have many beavers to catch 5 - 10 a year

Using a hooked utility blade its a lot easier to hook the jaw with the blade and go down to the vent from there

I think about odd things as well, but never gave this much thought. Since you asked, I think I do recall when I first started skinning beaver with a straight knife I went from the vent north. But when I first tried the hook knife it seemed logical to hook it in the lower lip and go south. Been doing it that way ever since. The only way I think it would matter is if one way is more likely to slice a castor, but I'm sure most have figured how to avoid that with their preferred method.

I never did join the hook blade crowd. Always been vent to snout with a sharp, pointy blade.

When using those hook blades you are cutting fur and skin.
Cutting from the arse up with the hook you will be cutting twice as much fur.
I use a knife with a small ball on the point to make the belly cut.Fast and clesn.

I use an old worn out Normark fillette knife to cut from the vent to the mouth....it seems to go easy enough, I'll not change anything now....unless that knife breaks.

The hooked blade Enters under skin and cuts just like a knife would. When I'm done I dom't see any fur laying around.
The only time you will see any cut fur Is when you cut across the fur when ringing the tail. And I don't see why that would really matter.

That doesn't make much sense to me.
Used that roofing blade for years, and never cut fur with it.
It's faster and cuts smoother by pulling than slicing.

Is the knife a special knife made just for you?
Never seen a knife with a ball for the tip.