In this paper, we investigate certain operational and inferential aspects of invariant PRAM (post randomization method) as a tool for disclosure limitation of categorical data. Invariant PRAMs preserve unbiasedness of certain estimators, but inflate their variances and distort other attributes. We introduce the concept of strongly invariant PRAM, which does not affect data utility or the properties of any statistical method. However, the procedure seems feasible in limited situations. We review methods for constructing invariant PRAM matrices and prove that a conditional approach, which can preserve the original data on any subset of variables, is an invariant PRAM. For multinomial sampling, we derive expressions for variance inflation due to invariant PRAMing and variances of certain estimators of the cell probabilities and also their tight upper bounds. We discuss estimation of these quantities and thereby assessing statistical efficiency loss due to invariant PRAMing. We find a connection between invariant PRAM and creating partially synthetic data using a nonparametric approach, and compare estimation variance under the two approaches. Finally, we discuss some aspects of invariant PRAM in a general survey context.