  Entanglement entropy may display a striking new symmetry under M\"obius
transformations. This symmetry was analysed in our previous work for the case
of a non-critical (gapped) free homogeneous fermionic chain invariant under
parity and charge conjugation. In the present work we extend and analyse this
new symmetry in several directions. First, we show that the above mentioned
symmetry also holds when parity and charge conjugation invariance are broken.
Second we extend this new symmetry to the case of critical (gapless) theories.
Our results are further supported by numerical analysis. For some particular
cases, analytical demonstrations show the validity of the extended symmetry. We
finally discuss the intriguing parallelism of this new symmetry and space-time
conformal transformations.
