DNA rearrangements such as sister chromatid exchanges (SCEs) are sensitive indicators of genomic stress and instability, but they are typically masked by single-cell sequencing techniques. We developed Strand-seq to independently sequence parental DNA template strands from single cells, making it possible to map SCEs at orders-of-magnitude greater resolution than was previously possible. On average, murine embryonic stem (mES) cells exhibit eight SCEs, which are detected at a resolution of up to 23 bp. Strikingly, Strand-seq of 62 single mES cells predicts that the mm9 mouse reference genome assembly contains at least 17 incorrectly oriented segments totaling nearly 1% of the genome. These misoriented contigs and fragments have persisted through several iterations of the mouse reference genome and have been difficult to detect using conventional sequencing techniques. The ability to map SCE events at high resolution and fine-tune reference genomes by Strand-seq dramatically expands the scope of single-cell sequencing.