**If it's already a vector shape**, cut it in half by deleting the anchors on one side, reflect a copy of what's left, and rejoin to make the sides symmetrical.

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If it's *not* a vector object.... redraw half the shape, tracing the original, then use the **Reflect Tool** and click the `Copy` Button. Then join the two halves.

[![enter image description here][1]][1]


To the best of my knowledge, there's no mechanism to symmetrically edit more than one existing anchor point, or existing Bezier handle. There are perhaps some third-party plug-ins which used to do that (i.e. Better Handles), but I don't know if they still exist for CC versions.


  [1]: https://i.sstatic.net/JPOnR.png