The ordering of morphemes in a word displays well-documented regularities across languages. Previous work has explained these in terms of notions such as semantic scope, relevance, and productivity. Here, we test a recently formulated processing theory of the ordering of linguistic units, the Efficient Tradeoff Hypothesis [Hahn et al., 2021]. The claim of the theory is that morpheme ordering can partly be explained by the optimization of a tradeoff between memory and surprisal. This claim has received initial empirical support from two languages. In this work, we test this idea more extensively using data from four additional agglutinative languages with significant amounts of morphology, and by considering nouns in addition to verbs. We find that the Efficient Tradeoff Hypothesis predicts ordering in most cases with high accuracy, and accounts for cross-linguistic regularities in noun and verb inflection. Our work adds to a growing body of work suggesting that many ordering properties of language arise from a pressure for efficient language processing.