  We consider central exclusive diffractive dipion production in the reactions
$pp \to pp \pi^{+} \pi^{-}$ and $p\bar{p} \to p\bar{p} \pi^{+} \pi^{-}$ at high
energies. We include the dipion continuum, the dominant scalar $f_{0}(500)$,
$f_{0}(980)$, and tensor $f_{2}(1270)$ resonances decaying into the $\pi^{+}
\pi^{-}$ pairs. The calculation is based on a tensor pomeron model and the
amplitudes for the processes are formulated in terms of vertices respecting the
standard crossing and charge-conjugation relations of Quantum Field Theory. The
formulae for the dipion continuum and tensor meson production are given here
for the first time. The theoretical results are compared with existing STAR,
CDF, CMS experimental data and predictions for planned or being carried out
experiments (ALICE, ATLAS) are presented. We show the influence of the
experimental cuts on the integrated cross section and on various differential
distributions for outgoing particles. Distributions in rapidities and
transverse momenta of outgoing protons and pions as well as correlations in
azimuthal angle between them are presented. We find that the relative
contribution of resonant $f_2(1270)$ and dipion continuum strongly depends on
the cut on proton transverse momenta or four-momentum transfer squared
$t_{1,2}$ which may explain some controversial observations made by different
ISR experiments in the past. The cuts may play then the role of a $\pi \pi$
resonance filter. We suggest some experimental analyses to fix model parameters
related to the pomeron-pomeron-$f_{2}$ coupling.
