Why kater's pendulum is called reversible pendulum ?

When calibrated, it has the same oscillation period when suspended from either set of pivots. The period is the same as an ideal simple pendulum ("ISP"; point m and mless string) of length equal to the distance between the pivot sets. This makes it much easier to determine g with high accuracy, because a true ISP cannot be built, and real-world approximations are very hard to build with sufficient precision to know the equivalent ISP length; also, the distance between the pivot sets can be very accurately measured. The calibration consists of adjusting a movable weight between the pivot sets until the periods are equal when the pendulum is suspended from either set. The ref. describes the theory.