What is Knitting into front and back of next stitch?

Knitting in the back is usually done to create a twist in the stitch.

This is often called for when picking up heel stitches while knitting a sock.

For comparison,

A regular knit stitch is usually done like this:

Put right-hand needle through the right-most stitch on the left needle, wrap yarn over right-hand needle, pull stitch through, and pull loop off the left-hand needle

A knit stitch through the back is done like this:

Put right-hand needle through the right-most stitch on the left needle on the back side of the needle (from the right, but catch the back side of the loop on the side of the needle that you can't see) , wrap yarn over right-hand needle, pull stitch through, and pull loop off the left-hand needle

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