A nice idea but not at all practical to do.  Nearly impossible, actually
as you would need to fuse a polarizing layer to the existing lens and
then, as you mentioned, stabilize the rotation.

A much better and easily accomplished solution is to wear polarized
sunglasses over regular contact lenses.

FYI, we do have tinted lenses in any shade or density and even
photochromic RGP lenses that get darker outside and clearer inside.

LB, O.D.

Yes, but unworkable for a couple of reasons.

"Polarized" lenses are a sandwich - two layers of plastic surrounding an
acetate film that is very similar to a photo negative. The acetate film
supports a layer of emulsion, a gel where silver halide crystals are formed
by preciptitation under an electric field.

The gel is probably permeable to oxygen when wet, but the acetate base film
would virtually block all oxygen from percolating through the contact into
the eye. A completely different carrier system would have to be developed.

The gel might be sandwiched into soft lens materials, but it would surely
swell and loosen when wet. It would be so fragile that rubbing and even
blinking would probably rearrange the gel, not to mention shaking up the
polarizing crystals. Then there are toxicity concerns since the materials
must be pretty incompatible with the eye, and would have to be sealed
between layers of (wet) contact lens material.

Toric contacts aren't 100% stable. It would be normal for the polarizing
axis to rotate a little, sometimes a lot.

All this for the limited benefit of polarizing, with the obvious drawback of
50% filter after sundown and screwey LCD instrument panels.

And the market is pretty limited, in part because those who really benefit
(fishermen, drivers) are arguably better off with spectacle lenses. Contacts
offer no impact protection, nor UV protection for the lids and adnexa, nor
can they be removed quickly.

-MT