Cancer is an evolutionary disease driven by mutations in asexually-reproducing somatic cells. In asexual microbes, bias reversals in the mutation spectrum can speed adaptation by increasing access to previously undersampled beneficial mutations. By analyzing tumors from 20 tissues, along with normal tissue and the germline, we demonstrate this effect in cancer. Non-hypermutated tumors reverse the germline mutation bias and have consistent spectra across tissues. These spectra changes carry the signature of hypoxia, and they facilitate positive selection in cancer genes. Hypermutated and non-hypermutated tumors thus acquire driver mutations differently: hypermutated tumors by higher mutation rates and non-hypermutated tumors by changing the mutation spectrum to reverse the germline mutation bias.