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In this paper we propose and extensively study mimetic $f({\cal G})$ modifiedgravity models, with various scenarios of cosmological evolution, with orwithout extra matter fluids. The easiest formulation is based on the use ofLagrange multiplier constraint.  In certain versions of this theory, it is possible to realize acceleratedexpansion of the Universe or even unified evolution which includes inflationwith dark energy, and at the same time in the same theoretical framework, darkmatter is described by the theory. This is achieved by the re-parametrizationof the metric tensor, which introduces a new degree of freedom in thecosmological equations and leads to appearance of the mimetic "dark matter"component. In the context of mimetic $f({\cal G})$ theory, we also provide somequite general reconstruction schemes, which enable us to find which $f({\calG})$ gravity generates a specific cosmological evolution. In addition, we alsoprovide the general reconstruction technique for Lagrange multiplier $f({\calG})$ gravity. All our results are accompanied by illustrative examples, withspecial emphasis on bouncing cosmologies.

Modified Gauss-Bonnet gravity with Lagrange multiplier constraint as mimetic theory