This paper gives an analytical characterization for the strategy-proof sequential allotment mechanisms of a limited resource that are equivalent to the straight and reverse priority mechanisms. The previously known equivalence of the anonymous priority mechanisms is extended to the non-anonymous case by establishing the following fact: for an arbitrary straight priority mechanism, there exists an equivalent reverse priority mechanism, but the converse fails. And finally, we provide a characterization for the class of reverse priority mechanisms having an equivalent straight priority mechanism.