If I select a node and want to scale both handles proportionately, I know I can do Shift+> or Shift+<.

Can I achieve the same result with the mouse? Moving/Scaling the handles simultaneously and proportionately?
Thanks!

If you click on the display handle's bounding box at the node tool's toolbar, then you can "rectangle select" each node's handles, and scale them with the mouse while holding Shift+Ctrl.

the "display handles bounding box at the node tool's toolbar" is the one that looks like four arrows pointing outwards? I've tried selecting and deselecting various buttons at the top but can't seem to be able to select multiple handles...

Yes, that is the right icon.

Rectangle select the node -click and drag the cursor from top left to bottom right-,
then click and drag on the handle's bounding box's corner handles, instead of the node's handles (and hold Shift+Ctrl).

After rectangle selecting, the transformation gizmo does not appear after selecting a single node, with the toolbar button selected or not.

I'll play with it and see if I can get it to show up, thanks for putting me on the right track!

Oh I forgot to mention that you have to enable the display at the inkscape preferences (Shift+Ctrl+P) under node/editing nodes of the right handles.

Ahhh there it is! Thanks so much!

Cheers!

If it's a smooth node, you can just change it to a symmetric node, and both handles move proportionally.