This paper describes the response of a superfluid neutron star to a sudden increase in the rotation rate of its crust - to a period jump - and shows how the mass, temperature, and distribution of density with radius within a pulsar can in principle be determined to great accuracy by observations of the subsequent spindown. Observations of successive jumps in the same pulsar can also yield general information concerning the mechanism producing the jumps. This program cannot now be realized. Detailed models are presented of a jump in the Crab pulsar, none of which successfully reproduce the observations. Qualitative reasons for this failure are given, and the particular developments, both observational and theoretical, that are required in order to complete the program are described.