  Late time cosmic acceleration may be achieved by modifying gravity on large
scales. This should also have consequences on the evolution of perturbations.
We thus extend our study of exponential infrared $f(T)$ teleparallel gravity to
examine the viability of the theory at the linear perturbation level,
evaluating the full CMB and matter power spectra. As the theory does not
introduce extra free parameters, it fits within the minimal six parameter space
of standard $\Lambda$CDM. Using Planck 2018 CMB (TT+TE+EE+lensing) alone, best
fits predict those parameters to be almost identical to $\Lambda$CDM, with
slightly smaller $\chi^2_{min}$. The resulting $H_0=72.24\pm 0.64$ km/s/Mpc,
which "practically" alleviates the tension with local measurements, due to late
time phantom behaviour. Inclusion of BAO data however reduces $H_0$, reflecting
furthermore systematic deviations from data that are also present in supernova
distances and the growth rate of structure (increasing the apparent tension in
the latter case). As the theory, unlike other viable $f(T)$ models, does not
reduce to $\Lambda$CDM through extra free parameters, those conclusions are
generic; applying to any modified gravity or dynamical dark energy with phantom
behaviour. With best fit parameters, the present scenario produces a CMB
spectrum almost identical to $\Lambda$CDM, with slight deviation at
low-multipole $\ell < 30$, where cosmic variance is large. The matter power
spectrum is also quite close to $\Lambda$CDM; with percent level scale free
modifications affecting modes significantly smaller than the horizon, arising
primarily from modified background evolution. More significant deviations
appear on larger scales, and may in principle distinguish modified gravity
scenarios of the type studied here from dynamical dark energy.
