We consider polynomial-time Turing machines which have access to two oracles and investigate when the order of oracle access is significant. The oracles used here are complete languages for the Polynomial Hierarchy (\PH). For language classes, we show that the order of the oracle queries does not matter, even when the queries are truth table queries. This improves upon the previous results of Hemaspaandra, Hemaspaandra and Hempel who showed that the order of the queries does not matter if the base machine asks one query to each oracle. For k > j, let H and E be complete languages for the k-th and j-th levels of PH, respectively. Then, our results show that for all polynomial bounded r(n) and s(n), languages recognizable by polynomial-time Turing machines which ask at most r(n) truth table queries to H followed by s(n) truth table queries to E can also be recognized by polynomial-time Turing machines that ask s(n) queries to E first, followed by r(n) queries to H. In fact, all of the queries can be made in parallel. On the other hand, we prove that, for computing functions, the order of oracle queries does matter, unless PH collapses. In particular, for all constants r and s, unless PH collapses to the (j+1)-th level, there exists a function that is computable by a polynomial time Turing machine which asks r truth-table queries to H followed by s truth-table queries to E that is not computable by any polynomial-time Turing machine which reverses the order of the queries to E and H. This talk presents joint work with Richard Beigel.