  We compute the first two k-invariants of the Picard spectra of $KU$ and $KO$
by analyzing their Picard groupoids and constructing their unit spectra as
global sections of sheaves on the category of manifolds. This allows us to
determine the E_\infty-structures of their truncations Pic(KU)[0,3] and
Pic(KO)[0,2]. It follows that these truncated Picard spaces represent: the
Brauer groups of Z/2-graded algebra bundles of Donovan-Karoubi, Moutuou and
Maycock; the Brauer groups of super 2-lines; and the K-theory twists of Freed,
Hopkins and Teleman. Our results also imply that that these spaces represent
twists of String and Spin structures on manifolds and can be used to twist
tmf-cohomology. Finally, we are able to identify pic(KU)[0,3] with a
cotruncation of the Anderson dual of the sphere spectrum.
