Pretty straightforward: use the Judoka, and NEVER TURN AROUND. Play on Day 1 for an easier time with it.
That's right, you'll only be facing to the right of the screen for an entire run.
The Judoka lets you toss enemies from your left side to your right. If you're lucky, you'll get a tile that grants you movement to the left, or a way to move enemies around. If you're unlucky, you can still work around it. Smoke Bomb is a great tile for this run, as is Swap Toss, Origin of Symmetry, Mirror and Knockback Punch.
Avoid Turn Around, for obvious reasons, unless you're going to turn around and sacrifice it for hella gold.