To recut worn checkering i use a Dem bart F1 fine single line cutter.You have to be careful,it may be better to practice on an old stock first. Use an old tootbrush to keep cutter cleaned out.Rabbit gave you some good info to follow. Bobby
What Bobby said. Either will work; I prefer 90 deg for everything. Install your cutter to cut on the pull stroke; it gives you much more control of where you're going. On the forward stroke you make a light crease that your cutter will follow on the pull stroke. Also, when pull cutting you don't have to charge the border. Cut a line to within 1/4" of the border, lift the cutter, carefully place it AT the border and pull back into the pattern. Do that religiously and you'll avoid a lot of overuns.
