Rotary press for treating hides, which comprises a support structure (10), a first and a second counter-rotating superimposed cylinder (2, 3), of which an upper cylinder (2) and a lower cylinder (3), which are rotatably mounted on the support structure (10). The press (1) further comprises at least one cushioning layer (4), which covers the lateral surface of at least one of the aforementioned upper and lower cylinders (2, 3), and such cushioning layer is made of a polymeric foam having a residual permanent deformation value after compression of less than 15%.