Palace of Darkness/No Logic
Kiki Skip is almost always the fastest way to get into Palace of Darkness and access everything but the boss. You can explore PoD in cave state, or get into Palace of Darkness state by leaving the dungeon, and holding down to bounce off the closed entrance and re-enter.
When you get to PoD via Kiki Skip, you will de-spawn all the enemies, statues, and crystal switches. You won't be able to change the state of the switch pegs, so you can't follow the path to the boss room. You don't need a bow for the eyegore room, because there are no eyegores. The bridge in the back tile won't collapse, because the invisible "overlord" sprite that makes it happen is gone as well.
You can't mirror in cave state, so you'll want to visit all the chests in a continuous route that includes backtracking on the right side.
If you fight Helmasaur, you'll probably get there with a glitch, because the intended route is very slow. Statue Drag to Helmasaur gets you there with minimal items. You could also use a lamp, bow, bombs, cane, and hookshot to hookpush through the north wall of the U-shaped rupee room into the Helmasaur arena.
Where you can go
Enter the door sideways with your sword out, and stop when the top 2 pixels of your sprite are visible.
Tap down and drink a red or blue potion at the same moment.
Pressing either up or down will cause you to enter the door immediately and get the wrong effect, so press left or right until you're a few pixels out of the door. Then press down to transition through the door. You end up a subtile to the north and then autowalk to a valid location.
The important thing is that you've loaded (from the room below) the ability to fall into a different room (a wrong warp), and in this location that happens to be the Tower of Hera. Walk off the ledge to get there.