  The two-dimensional color code is an alternative to the toric code that
encodes more logical qubits while maintaining crucial features of the
$\mathbb{Z}_2\times\mathbb{Z}_2$ toric code in the long wavelength limit.
However its short range physics include single qubit Pauli operations that
violate either three or six stabilisers as opposed to the toric code where
single qubit Pauli operations violate two or four stabilisers. Exploiting this
fact we construct several non-CSS versions of the two-dimensional color code
falling into two families - those where either three, four or five stabilisers
are violated and those which violate exactly four stabilisers for all the three
types of single qubit Pauli operations. These models are not equivalent to the
original color code by a local unitary transformation. Nevertheless the code
spaces of the CSS and non-CSS versions are related by local unitaries which
reflects the fact that their long range physics coincide. This also implies
that the non-CSS versions host transversal Clifford gates and hence support
fault-tolerant computations. As a consequence of the non-CSS structure, the
logical operators are of a mixed type which in some cases include all the three
Pauli operators making them potentially useful for protection against biased
Pauli noise.
