I think the comments answered the question, but it seems like some pictures would be helpful as well. There are many examples, but one of my favorites is [this][1] paper by Sakuma which concludes with a table of knots preserved by orientation-preserving involutions on $S^3$ which reverse the orientation on $K$.

I also want to point out that in general studying prime order group actions is not sufficient. For example, you may ask if the figure eight knot $4_1$ is preserved by a symmetry which reverses the orientation on $S^3$, but preserves the orientation on $4_1$. This is the case, but the symmetry has order 4 (and there is no such order 2 symmetry). Visually, this can be seen in the following image as a $\pi/2$ rotation within the plane of the diagram followed by a reflection across an $S^2$ intersecting the diagram in the shown dotted green circle.[![Figure eight knot symmetry][2]][2]


  [1]: http://www.math.sci.hiroshima-u.ac.jp/sakuma/papers/StronglyInvertibleKnots.pdf
  [2]: https://i.sstatic.net/mFj2M.png