  We define a turning of a rank-$2k$ vector bundle $E \to B$ to be a homotopy
of bundle automorphisms $\psi_t$ from $\mathbb{Id}_E$, the identity of $E$, to
$-\mathbb{Id}_E$, minus the identity, and call a pair $(E, \psi_t)$ a turned
bundle. We investigate when vector bundles admit turnings and develop the
theory of turnings and their obstructions. In particular, we determine which
rank-$2k$ bundles over the $2k$-sphere are turnable.
  If a bundle is turnable, then it is orientable. In the other direction,
complex bundles are turned bundles and for bundles over finite $CW$-complexes
with rank in the stable range, Bott's proof of his periodicity theorem shows
that a turning of $E$ defines a homotopy class of complex structure on $E$. On
the other hand, we give examples of rank-$2k$ bundles over $2k$-dimensional
spaces, including the tangent bundles of some $2k$-manifolds, which are
turnable but do not admit a complex structure. Hence turned bundles can be
viewed as a generalisation of complex bundles.
  We also generalise the definition of turning to other settings, including
other paths of automorphisms, and we relate the generalised turnability of
vector bundles to the topology of their gauge groups and the computation of
certain Samelson products.
