Here's the controls I discovered through trial and error:
Left Arrow, A, or H to move left.
Right Arrow, E, D, or L to move right.
Up Arrow, W, K, or Comma (,) to enter doors or jump.
Hold the Down arrow, Shift, Ctrl, S or J to sneak (move at about 50% speed).
R to reset room (perhaps in case you get softlocked and can't die).
T to toggle "Deathhead switch" (a number floating above your head showing how many times you've died)
Z to toggle "Bunnyhop switch" (whether holding a jump button jumps repeatedly or just once)
The Enter key toggles Fullscreen/Windowed
VERY IMPORTANT: You will move about twice as fast and be able to jump once you start the game and go through a couple of doors. Navigating menus and settings is the same as walking in-game, and is painfully slow if you try to do it before actually starting the game.
Esc key opens pause "menu" (Exit, Settings, Continue)
Jumping over to the left from here also lets you access the Language Menu (German/English available, Spanish and French visible but inaccessible) as well as a link to the website and an "Erase Savestate" option (which takes effect immediately when you enter that door). Up to the right is an inaccessible "Hardmode 10 lives" door.
Settings Menu options from left to right:
Borderless Switch, 960-540, 576-324, 2560-1440, 1152-648, 1600-900, 1920-1080, Fullscreen Switch, (settings menu spawnpoint), Back, Music 100/0 Percent, Sound 100/0 Percent, Deathhead Switch, Bunnyhop Switch.
None of the resolution, fullscreen or sound options are preserved after exiting the game, it will always launch at 1152x648 windowed with 100% music and sound.
Mouse controls: left-click to move left, right-click to move right, middle-click to jump or enter doors
Dual-Shock 4 controls:
Dpad, Left analog stick or right analog stick to move left and right.
X button, L1 or R2 to jump/go in doors.
O button or L2 to sneak.
Options button for pause "menu".
I don't have an Xbox controller to test, but I assume it'd be similar like A/LB/RT ; B/LT ; Start