Photoshop doesn't have such a function natively, but you can fake it, if a little awkwardly.
Start your shape. Select the shape layer AND the background layer and turn them into a Smart Object (Filter > Convert for Smart Filters or Layer > Smart Objects > Convert to Smart Object). I'm assuming you'll make this the left hand half, so position it to the left of the document. (For other relationships adjust these steps accordingly.)
Copy the layer by pressing Cmd/Ctl-J or by dragging it to the New Layer icon. You now have a copy of the same shape and background layer.
Double-click either Smart Object to open it as its own document.
Press Cmd/Ctl-T to activate Free Transform on your shape layer. In the transform proxy in the control panel, change the transform position indicator from the center (default) to the right-hand middle position.
Right-click the object and select "Flip Horizontal." Press Enter/Return to lock in the transform.
Choose Window > Arrange > Tile.
You now have your main canvas and the master shape visible. Any changes you make to the master shape will appear in the main document every time you save the .psb document that is your smart object.
The plus with this approach is that all the tools of Photoshop are at your disposal in working on one half of the shape, and will be immediately duplicated in the other half as you save.
The downside is that creating a merged, single shape requires several extra steps. I'll outline them if you need to do that.
This is also the best way to make reflected text in a Photoshop composition and keep the text editable, by the way. Smart Objects opened up a whole new realm of possibility when they were introduced.