Check out Mobius transformations (also known as linear fractional transformations) of the complex plane. They reflect the entire complex plane across a circle (or a line, which is a circle with its center at infinity). This is done conformally (in a way that preserves angles, so that locally it looks like you're only scaling, rotating, and translating), which I think you want; otherwise, the image of your curve might not bear any resemblance whatsoever to what you started with.

The bad news is that if you need to conformally map the entire plane onto itself (e.g. one side of a curve to the other side, and conversely), then Mobius transformations are all you have to work with.