How do I rebind the controls in BoI: Rebirth?

Yes, you can rebind the controls. From the main menu, choose "options"

Main Menu

Then choose "controls"

Options

Then choose "keyboard"

Controls

Then you will see this screen

Keyboard controls

From here, you can move between the options with the arrow keys, select by pressing Space, and choose a different key by pressing that key.

In order to change the controls mid-run, exit the game to auto-save your progress. Then follow the steps as before and from the main menu, choose "continue".