These levels can only be accessed by glitching through a specific staircase in the Castle Axis chapter. The easter egg is the first two levels from the very first chapter of the game, but with the background and level geometry all bugged out.
The levels can still be beaten, but they are significantly harder since you can't tell where you're going at all and need to use your memory of the levels in order to clear them.
Unfortunately, the third level is inaccessible, as once you clear the second level, you are met with a kill screen (the last image in the post). This means that it is impossible to clear the entire chapter or play the rest of the game with this easter egg active, which is a real shame 
I made a post about this discovery exactly 2 months ago, which the developer also responded to.
Here's the link to the initial post: https://gamejolt.com/p/secret-levels-found-in-swordswinger-gj-omg-5njsjtkq
I think I may just be the very first person ever to find this easter egg, which is such a cool thing to think about, even if the easter egg is from a very niche and unknown game.
I hope @efefski 
updates this easter egg if he chooses to revisit Swordswinger again in the future. I'd love to see if I could beat the whole game this way, I'm always up for a challenge 
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