  Starting from the defining two-point and three-point functions of Celestial
CFTs, Euclidean integral blocks are constructed for the OPE of scalar
primaries. In their integral form they can alternatively be fixed using
Poincar\'e symmetry acting on both massless and massive states. Subsequently,
an analytic continuation is done to define the Lorentzian version of the
correlation functions and the OPE blocks as valued on the $(1,1)$ cylinder, the
universal cover of the recently studied celestial torus. The continuation is
essentially the same that is used in the derivation of the KLT relations for
string amplitudes. It is shown explicitly that the continued OPE blocks encode
the contributions from massive primary states as well as their shadow and
light-transformed partners of continuous spin. The corresponding pairings are
also studied, thus the construction provides the fundamental relation between
correlation functions and OPE coefficients in the scalar CCFT.
