  In the presence of a nonzero B-field, the symmetries of the $\mathrm
E_8\times \mathrm E_8$ heterotic string form a 2-group, or a categorified
group, as do the symmetries of the CHL string. We express the bordism groups of
the corresponding tangential structures as twisted string bordism groups, then
compute them through dimension 11 modulo a few unresolved ambiguities. Then, we
use these bordism groups to study anomalies and defects for these two string
theories.
