  Customarily, in-plane auxeticity and synclastic bending behavior (i.e.
out-of-plane auxeticity) are not independent, being the latter a manifestation
of the former. Basically, this is a feature of three-dimensional bodies. At
variance, two-dimensional bodies have more freedom to deform than
three-dimensional ones. Here, we exploit this peculiarity and propose a
two-dimensional honeycomb structure with out-of-plane auxetic behavior opposite
to the in-plane one. With a suitable choice of the lattice constitutive
parameters, in its continuum description such a structure can achieve the whole
range of values for the bending Poisson coefficient, while retaining a
membranal Poisson coefficient equal to 1. In particular, this structure can
reach the extreme values, $-1$ and $+1$, of the bending Poisson coefficient.
Analytical calculations are supported by numerical simulations, showing the
accuracy of the continuum formulas in predicting the response of the discrete
structure.
