Thanks for the reply! I guess where I’m still puzzled is that–even on the view you outline–the sense in which a vector has an orientation seems to be completely disjoint from the way a hypervolume gets its orientation.
Rather consider the sign and the scalar separately.
Well, the way the aforementioned signs and scalars are calculated is so radically different for vectors than it is for hyper volumes, that it seems to me that they should even have different names.
To emphasize the point, for the vanilla algebras which a newb would be familiar with, the sign of the length of a vector can never be less than zero. So we say that all nonzero vectors always have the same orientation? In vanilla algebras, even multiplying a vector by 1 won’t change its length. Surely this is completely different for hyper volumes?
And, as pointed out above, it’s computed with a quadratic function for a vector, which is ipso factor nonlinear.
cf. with the orientation of a bivector or higher volume:

They don’t all share the same orientation, like vectors (apparently?) mostly do.

Multiplying a hypervolume by 1 will flip the orientation.

And for hypervolumes, it is a bi or multilinear function.
I guess the most succinct way of putting my point is that for vectors, the length is calculated from the dotproduct part of the geometric product, while volume is calculated by the wedgeproduct part of the geometric product. Completely disjoint parts of the geometric product—why is the same concept of orientation given to both? Seems like equivocation.
Once you square an entity, the resulting value is no longer of the same type
Absolutely. There is a very fundamental and deep distinction between length and volume. That’s why bivectors can’t be vectors (despite what that Yankee Dr. Gibbs says).
But this doesn’t help me understand why we lump the orientation of vectors in
conceptually with the orientation of columns: in fact, it just underscores how weird it is to do that.
I mean, this might seem like a small or pedantic point, but virtually every GA tutorial tries to build intuition for why volumes should be oriented by comparing/analogising/equivocating the orientation of vectors with the orientation of blades. This really tripped me up.
Seems like either:

A newb reading such a tutorial would just bleep over the intro to orientation, in which case the comparison is doing no good, or,

A newb like me, reading a tutorial like that, would pause to ponder it over a bit–and the more they thought about it the more mysterious it would appear. Points of preoccupation would be, in rough chronological order:

Hey wait, the magnitude of a vector is always positive (newbs haven’t learned about various different metrics yet) so how does it make sense even to say a vector is oriented?

But this tutorial says vectors have orientation, so what is the orientation of a vector? It can’t be the (positive) length of the vector…but then what the heck is it?

I can see how a bivector gets its orientation a^b = b^a, So it would seem natural that the anticommutativity of the wedge product would have some geometric meaning, And its straightforward that there are two ways–clockwise and counterclockwise–to sweep one vector to another. But if we are trying to do “the same thing” for vectors, we have a problem: there’s just one vector, what are we sweeping?

Hey wait, the function to calculate the length of a vector is not even linear, whereas calculating oriented hyper volumes is a linear function. The distinction between linear and nonlinear in functions is a pretty big distinction, so why isn’t that distinction preserved in the terminology, like saying vectors don’t have orientation?

and (thanks to my conversational partner!) The units don’t even make sense. The length of a line is in meters, and the area of a bivector is in m^2.
Instead of a nice, smooth on ramp, the road just kind of breaks up into little chunks of concrete, and the further I go down it, the less of the road is left leaving me in the weeds
Some conceptual help here would be very appreciated!! What is the orientation of a vector, and how do we compute it? And why does it make sense to lump:
 The square root of a dotproduct, and
 The wedge product
into the same category? Why would you use the same word, “orientation”, for both?