  Taking $\mathbf{O}(D,D)$ covariant field variables as its truly fundamental
constituents, Double Field Theory can accommodate not only conventional
supergravity but also non-Riemannian gravities that may be classified by two
non-negative integers, $(n,\bar{n})$. Such non-Riemannian backgrounds render a
propagating string chiral and anti-chiral over $n$ and $\bar{n}$ dimensions
respectively. Examples include, but are not limited to, Newton--Cartan,
Carroll, or Gomis--Ooguri. Here we analyze the variational principle with care
for a generic $(n,\bar{n})$ non-Riemannian sector. We recognize a nontrivial
subtlety for ${n\bar{n}\neq 0}$ that infinitesimal variations generically
include those which change $(n,\bar{n})$. This seems to suggest that the
various non-Riemannian gravities should better be identified as different
solution sectors of Double Field Theory rather than viewed as independent
theories. Separate verification of our results as string worldsheet
beta-functions may enlarge the scope of the string landscape far beyond
Riemann.
