The pseudosphere or tractroid surface, the surface of revolution based on the tractrix, is supposed to be a surface of constant negative curvature.
This means the product of the two principal curvatures is constant, so they must be inverse of each other.
Since the tractroid is a surface of revolution, the curvature of its circular cross sections is the inverse of the circles' radii. The radii of the circles is simply x (for a tractrix with a vertical asymptote), so their curvature is 1/x.
The curvature along the curves orthogonal to the circles must be the inverse of the curvature of the circles, or x.

By this logic, the curvature of the tractrix must be x. Where am I going wrong?

Thanks

It is.

>This means the product of the two principal curvatures is
>constant, so they must be inverse of each other.

With sign minus.

>Since the tractroid is a surface of revolution, the
>curvature of its circular cross sections is the inverse of
>the circles' radii. The radii of the circles is simply x
>(for a tractrix with a vertical asymptote), so their
>curvature is 1/x.

Right.

>The curvature along the curves orthogonal to the circles
>must be the inverse of the curvature of the circles, or x.

At best, -x, right?

>By this logic, the curvature of the tractrix must be x.
>Where am I going wrong?

The tractrix does not have a constant curvature; so you know that somewhere there is a flaw in your reasoning.

I think that the starting point may be in the realization that you should be dealing with local coordinates on the surface. If you imagine those, then the second realization may be that the circle you are talking about (the one of radius x) is not exactly the circle whose radius directly leads to one of the principal curvatures. I'd say that the right circle is bound to be slanted.