  In this paper we propose and extensively study mimetic $f({\cal G})$ modified
gravity models, with various scenarios of cosmological evolution, with or
without extra matter fluids. The easiest formulation is based on the use of
Lagrange multiplier constraint.
  In certain versions of this theory, it is possible to realize accelerated
expansion of the Universe or even unified evolution which includes inflation
with dark energy, and at the same time in the same theoretical framework, dark
matter is described by the theory. This is achieved by the re-parametrization
of the metric tensor, which introduces a new degree of freedom in the
cosmological equations and leads to appearance of the mimetic "dark matter"
component. In the context of mimetic $f({\cal G})$ theory, we also provide some
quite general reconstruction schemes, which enable us to find which $f({\cal
G})$ gravity generates a specific cosmological evolution. In addition, we also
provide the general reconstruction technique for Lagrange multiplier $f({\cal
G})$ gravity. All our results are accompanied by illustrative examples, with
special emphasis on bouncing cosmologies.
