  In this article we study in detail the category of noncommutative motives of
separable algebras Sep(k) over a base field k. We start by constructing four
different models of the full subcategory of commutative separable algebras
CSep(k). Making use of these models, we then explain how the category Sep(k)
can be described as a "fibered Z-order" over CSep(k). This viewpoint leads to
several computations and structural properties of the category Sep(k). For
example, we obtain a complete dictionary between directs sums of noncommutative
motives of central simple algebras (=CSA) and sequences of elements in the
Brauer group of k. As a first application, we establish two families of motivic
relations between CSA which hold for every additive invariant (e.g. algebraic
K-theory, cyclic homology, and topological Hochschild homology). As a second
application, we compute the additive invariants of twisted flag varieties using
solely the Brauer classes of the corresponding CSA. Along the way, we
categorify the cyclic sieving phenomenon and compute the (rational)
noncommutative motives of purely inseparable field extensions and of dg Azumaya
algebras.
