If you have never heard of a kaschtoir, you are not alone. The term is a portmanteau coined by Peter Verheyen. The kaschtoir is a backing tool; a combination of a German kaschiereisen and a French frottoir. More about the kaschiereisen here, and more about the frottoir here. This tool combines the most useful aspects of each.
The kaschiereisen end has small teeth, which help to move the sewn, rounded bookblock into to a backed position. The froittoir end has a gentle smooth curve, which also can help move a bookblock into position, in addition to smoothing out irregularities and trueing raised bands. Two for the price of one!
If you are tired of deforming your spine into double folds with a hammer, or deforming your fingers hand manipulating your spine into shape, this may be the tool for you. Fits comfortably in one or two hands. The stainless steel is safe for contact even with historic bookblocks. The edges are very comfortably rounded, and the froittoir end is polished to make clean up easy.
I reproduced the gentle curve on the Frottoir from examples in my historic collection. This tool is not a die meant to exactly shape a spine to its curvature. A gentle curve is much more useful for a range of round spine shapes, smoothing irregularities, smushing sewing threads, etc… . The teeth can be cleaned of adhesive with a stiff brush. This tool is quite heavy, and the weight allows you to easily persuade even hard modern paper into the shape you desire with little effort.
304 stainless steel. Approximately 6 x 2 x .5 inch (15 x 5 x 1.25cm). Weighs about 1 lb. 5 oz (600 grams).