Nematicity in superconductors

it seems to me that the definition of "nematic phase" is quite chamaleonic.

Quoting from its first definition [Nature 393, 550 (1998)] for a 2-dimensional square lattice "The nematic phase breaks the four-fold rotation symmetry of the lattice, but leaves both translation and reflection symmetries unbroken". This is clearly not possible; if you don’t eliminate even reflection symmetries, you cannot break the 4-fold rotation symmetry (mirror planes act as generators of the 4-fold rotation symmetry). Then it is not clear if the therm ‘lattice’ refers to a true mathematical ‘lattice’ or to the crystal structure (that is a convolution of the lattice with the basis).

Quoting from [Phys. Rev B77, 224509 (2008)] in iron superconductors "… a closely associated structural transition, which we wish to identify as the transition to an “electron nematic phase”". But the structural transition is from the tetragonal to the orthorhombic system and actually breaks both mirror and translation symmetries. It seems to me that this statement is in contradiction with the above definition.

in another paper it is stated "…the distortion is driven by electronic nematicity rather than a lattice (phonon) instability" [Nature Physics 11, 959–963(2015)].

So, is it a nematic transition a structural transition simply not driven by lattice (structural) degrees of freedom?