I spoke to Tony Milton, who says he’s seen people use convex whetstones in Japan - for knives - but a convex strip was a new one on him, too. I knew it wasn’t defective - it’s a hand-made Inoue Tōsake item, initialled in gold pen on the end (I don’t know what the kanji are; they look similar to the Tōsake kanji on the box, but I’m far from an expert!).
The dealer simply said that the convex side was intended to provide a more delicate stropping effect, specifically for polishing the edges of kamisori blades. Considering the geometry, it made sense that the effect would be slightly different, but in the end I decided the only way to check was to test it out.
10 laps on stone, 40 on canvas, then 100 on the straight edge for shave 1; the exact same for shave 2 except the 100 laps on leather were divided 50/50 between the flat and the convex strop.
The results? They both have blindingly good shaves - one pass, followed by a partial pass along the neck area. On shave 1 I did give the blade 10 laps on canvas after the initial pass, whereas I didn’t feel the need on shave 2. Tbh, the difference is negligible at most, but one thing is clear - it’s a perfect strop for a kamisori user.