In the presence of a nonzero B-field, the symmetries of the
E
8
×
E
8
\mathrm E_8\times \mathrm E_8
heterotic string form a
2
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
11
modulo a few unresolved ambiguities. Then, we use these bordism groups to study anomalies and defects for these two string theories.