  We prove a complexity dichotomy theorem for symmetric complex-weighted
Boolean #CSP when the constraint graph of the input must be planar. The
problems that are #P-hard over general graphs but tractable over planar graphs
are precisely those with a holographic reduction to matchgates. This
generalizes a theorem of Cai, Lu, and Xia for the case of real weights. We also
obtain a dichotomy theorem for a symmetric arity 4 signature with complex
weights in the planar Holant framework, which we use in the proof of our #CSP
dichotomy. In particular, we reduce the problem of evaluating the Tutte
polynomial of a planar graph at the point (3,3) to counting the number of
Eulerian orientations over planar 4-regular graphs to show the latter is
#P-hard. This strengthens a theorem by Huang and Lu to the planar setting. Our
proof techniques combine new ideas with refinements and extensions of existing
techniques. These include planar pairings, the recursive unary construction,
the anti-gadget technique, and pinning in the Hadamard basis.
