  The Checkerboard conformal field theory is an interesting representative of a
large class of non-unitary, logarithmic Fishnet CFTs (FCFT) in arbitrary
dimension which have been intensively studied in the last years. Its planar
Feynman graphs have the structure of a regular square lattice with checkerboard
colouring. Such graphs are integrable since each coloured cell of the lattice
is equal to an R-matrix in the principal series representations of the
conformal group. We compute perturbatively and numerically the anomalous
dimension of the shortest single-trace operator in two reductions of the
Checkerboard CFT: the first one corresponds to the Fishnet limit of the twisted
ABJM theory in 3D, whereas the spectrum in the second, 2D reduction contains
the energy of the BFKL Pomeron. We derive an analytic expression for the
Checkerboard analogues of Basso--Dixon 4-point functions, as well as for the
class of Diamond-type 4-point graphs with disc topology. The properties of the
latter are studied in terms of OPE for operators with open indices. We prove
that the spectrum of the theory receives corrections only at even orders in the
loop expansion and we conjecture such a modification of Checkerboard CFT where
quantum corrections occur only with a given periodicity in the loop order.
