Hmmmm.
It is called a shear web, because its primary function is to assist the two spar caps such that all three elements behave as a single structural piece. The webbing in fact, carries shear loads. The shear vector is parallel with the spar, ie. spanwise. The peak shear value is at the bending neutral axis which is approximately the vertical center, assuming that the spar has top/bottom symmetry. If this concept is not obvious, think of it this way: in a hard inside pull-up, the wing is loaded such that the upper spar cap is in compression, and the bottom cap is in tension. The web has to connect both of these, whose bending stresses go thru a change of sign. Hint: the web does it by carrying shear loads.
Tim is correct that if you think of two spar caps behaving independently (ie. two separate beams) that adding a web allows both a greater load without buckling. But it is confusing to say that the only purpose of the shear web is to prevent buckling. That's not correct.
As everyone intuits, wood of any kind does not have isotropic properties. You want to shear balsa across the grain, hence, the vertical orientation. That orientation is also preferred for compressive stress between the caps. The same is true for a TE "beam" but I suspect that the section depth is small enough (1/4" to maybe 3/8" for most models?) and the wood used on a TE (generally 1/16") is not real robust, so using span-wise grain balsa probably helps stiffness, and doesn't hurt it. The issue there, though, is that at each rib there is a discontinuity, so even with good fits, you're not gaining as much as you might think. The discontinuity is that the rib grain orientation does not match the lateral web grain, and it would be subject to crushing. A vertical grain orientation would reduce this tendency.
You may not want a super rigid wing. If the plane is light, and you fly over grass, and you can let the structure flex to absorb the energy, it is more likely to rebound without damage. If you web it all up, and the structure cannot deflect without rupturing, then the structure explodes.
I would agree with Ken's statement if we always built the same size, same weight, same same kind of planes. Which we mostly do, so his rules of thumb probably work nearly always. But if you understand the load path, and where the stresses concentrate, and look at construction options, you actually do have choices. In a lot of cases we are overbuilt until it hits the ground. So use a construction technique that is easy for your building tools/skills/style.
A good resource to explain this if anyone is trying to understand this is a college Strength of Materials text such as Higdon.
Dave