In many of my designs, I use K2TOG (knit 2 together) and SSK (slip, slip, knit) to join live stitches perpendicularly. The photos below illustrate the beginning and ending of each RS (right side) and WS (wrong side) row using this join-as-you-go knitting technique.
The example photos show live stitches (green) on the needle being joined to live stitches on an adjacent perpendicular piece (pink). In some of my patterns, I join live stitches on the needle to selvedge stitches of adjacent pieces. The only difference between using a live stitch and a selvedge stitch is how the selvedge stitch gets onto the needle. Always pick up both legs of the selvedge stitch from WS to RS, and then work K2TOG or SSK (depending on whether you are at the start or end of the row) as directed.