If I start with an infinite flat sheet of graph paper, and in polar coordinates cut out a piece according to: $r>0, \ \ -f(r) < \theta < f(r)$
Now I want to stitch the remaining graph paper together, by associating each point $(r,f(r))$ to $(r,-f(r))$ on the seam.
How do I calculate what the curvature is of the stitched up paper along the seam?
Perhaps your description of what you wanted to do doesn't match with what you intended. If we go backwards, i.e., start with a right circular cone, and make some squiggly cut in it, from base to apex, and then flatten that out we will get a disk with a wedge with squiggly sides cut out. The two sguiggly sides match up.
Assume that if we did the cut with a straight line, then we'd get a wedge described by the lines $\pm \tilde{\theta}$. Now, we can imagine replacing those straight lines by the squiggly one described by your function. To keep from making a mess, we'd need constraints on the function, corresponding to your squiggly line on the cone not wrapping around and intersecting itself. Say $f'(0) = 0$, and $|f(r)| < \pi r$, but we might have to think about that more. Now we have a wedge cut out by curves described by $\pm \tilde{\theta} + f(r)$. In other words, we're removing the piece according to $$ -\tilde{\theta} + f(r) \leq \theta \leq \tilde{\theta} + f(r), $$ which is not the same as you have in your question. If we get it right, and we have a seam with no curvature, then it will be represented by a curve on the cone we get, so it makes sense to go backwards like this.