After you turn your outside corner, you will work standard buttonhole stitches usually until you reach the end of a Kloster. The down hole of your buttonhole stitch will share a corner with the Kloster (see picture). The down hole of your next buttonhole stitch will share a corner with the perpenicular Kloster (my needle is in the shared corner of the second Kloster). The inside corner of the buttonhole is formed when the "up" hole is shared. In the picture, my needle is coming up in the loop formed from the previous up hole. It's a good idea to keep your thumb in the loop while you place your needle, turn the fabric, and bring the needle up. Then pull it taught as you pull your working thread through.
Catching this loop is essential to maintaining that ridged edge. When I learned this technique, it wasn't explained very well, and I had to figure it out on my own. You will get so that you will be able to effortlessly place, turn and pull on these inside corners.
By the way, these inside corners are the very best places to end your thread, because there's a natural break in the flow, and it isn't as obvious.
You can look at my "Baroque" piece for an example of some of the complex shapes you can create - there are a lot of inside corners on that one! You will also use this a lot if you are outlining simple diamond shapes.