Recent studies have shown that language models pretrained and/or fine-tuned on randomly permuted sentences exhibit competitive performance on GLUE, putting into question the importance of word order information. Somewhat counter-intuitively, some of these studies also report that position embeddings appear to be crucial for models' good performance with shuffled text. We probe these language models for word order information and investigate what position embeddings learned from shuffled text encode, showing that these models retain information pertaining to the original, naturalistic word order. We show this is in part due to a subtlety in how shuffling is implemented in previous work - before rather than after subword segmentation. Surprisingly, we find even Language models trained on text shuffled after subword segmentation retain some semblance of information about word order because of the statistical dependencies between sentence length and unigram probabilities. Finally, we show that beyond GLUE, a variety of language understanding tasks do require word order information, often to an extent that cannot be learned through fine-tuning.