I have noticed for some time that I solve F2L differently than pretty much everyone, so I want to share my F2L method and get some inputs about this way of doing F2L. I also would want to know if this has ever been done before.
I called my method Sideways F2L. Sideways F2L starts after finishing a cross like a regular CFOP F2L, but instead of holding the cube with yellow on top, I would hold my cube sideways so the (white) cross is on the left and last layer (yellow) is on the right (reverse this if you're a leftie). I'll then be pairing and inserting using mostly just l/l', U/U', and R/R'.
Pairing is done intuitively, and is pretty simple as there are significantly less cases than with standard F2L as we can eliminate mirrors and backsides with l and R rotations.
There are two basic insertion algorithms, which I call Thumb Insert (U' R U) and the Pinkie Insert (U R' U') with l or R rotation to choose the side to insert to. Pretty much any pairs can be inserted efficiently to any positions with just those two insertion algorithms. There are also Sledgehammer variants for both Thumb and Pinkie pair inserts, which are useful for LL edge control.
Some advantages I noticed of Sideways F2L is that it is easy to learn intuitively while allowing high speed pairings and insertions. Since Sideways F2L uses l/l' instead of Y/Y', you never need to do two hands Y/Y' regrips, so you don't need a separate rotationless algorithms to avoid clunky Y/Y' regrips. Also, most of the work is done with your dominant hand so that's a plus if you have a strong preference on your handedness. Doing lookahead is also not too difficult, but it requires a somewhat different lookahead techniques than standard F2L.
I usually also do Sideways Cross, which transitions nicely to Sideways F2L, though I've never been that good at neither regular nor sideways cross. When finishing F2L, I usually insert in the last pair using fingertrick variants that leave the last layer on top, and then just do a standard last layer. Recently, I've also been incorporating edge control for inserting the last pair, which works very well because most Sideways Sledgehammer insertions have fingertrick variants which will also rotate the cube to leave the last layer on top.
I always average about 5-10 seconds faster using Sideways F2L than with regular F2L, even though I practice both roughly equally (my current average is 30s). So it works well for me so far, though since most of the very fast people are using standard F2L, I've also been wondering if I would benefit from focusing on just one way to do F2L rather than trying to practice both.
