We present new multiband (UBVI) time-series data of helium burning variables in the Carina dwarf spheroidal galaxy. The current sample includes 92 RR Lyrae-six of them are new identifications-and 20 Anomalous Cepheids, one of which is new identification. The analysis of the Bailey diagram shows that the luminosity amplitude of the first overtone component in double-mode variables is located along the long-period tail of regular first overtone variables, while the fundamental component is located along the short-period tail of regular fundamental variables. This evidence further supports the transitional nature of these objects. Moreover, the distribution of Carina double-mode variables in the Petersen diagram (P-1/P-0 versus P-0) is similar to metal-poor globulars (M15, M68), to the dwarf spheroidal Draco, and to the Galactic Halo. This suggests that the Carina old stellar population is metal-poor and affected by a small spread in metallicity. We use trigonometric parallaxes for five field RR Lyrae stars to provide an independent estimate of the Carina distance using the observed reddening free Period-Wesenheit [PW, (BV)] relation. Theory and observations indicate that this diagnostic is independent of metallicity. We found a true distance modulus of mu = 20.01 +/- 0.02 (standard error of the mean) +/- 0.05 (standard deviation) mag. We also provided independent estimates of the Carina true distance modulus using four predicted PW relations (BV, BI, VI, BVI) and we found: mu = (20.08 +/- 0.007 +/- 0.07) mag, mu = (20.06 +/- 0.006 +/- 0.06) mag, mu = (20.07 +/- 0.008 +/- 0.08) mag, and mu = (20.06 +/- 0.006 +/- 0.06) mag. Finally, we identified more than 100 new SX Phoenicis stars that together with those already known in the literature (340) make Carina a fundamental laboratory for constraining the evolutionary and pulsation properties of these transitional variables.