Given a finite graph $G$, we characterize the automorphisms and automorphism groups of $G$ that can be realized as geometric symmetries and symmetry groups of straight line drawings of $G$ in the plane. In doing so, we correct a previous version of this characterization from the literature. As an application, we show that the graphs recently constructed by the first two authors to disprove the Polycirculant Conjecture, which have orders $16{,}464^k$ for each positive integer $k$, are the first known vertex-transitive graphs with no non-trivial realizable geometric symmetries. We also construct infinite families of graphs that exhibit all possible kinds of geometric symmetry.

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