Semi-regular variables (SRVs) are similar to Miras in brightness, and they also follow PLRs, though not necessarily the same as Miras. As potential standard candles they are more challenging than Miras due to their smaller variability amplitudes and less regular light curves, but they are substantially more numerous and especially promising to probe old stellar populations. We aim to characterize the variability of SRVs, with focus on their connection with Miras, in order to prepare the ground for investigating their potential as distance indicators. We examine SRVs and Miras in the Magellanic Clouds from OGLE-III observations, with data from Gaia and 2MASS. After cleaning the sample from variability periods unrelated to pulsation, we classify each source by chemical type and combination of pulsation modes. We examine the results in terms of global photometric and pulsation properties. We identify four SRVs groups that fit the general evolutionary scenario predicted by theory. SRVs dominated by fundamental-mode pulsation are very similar to Miras, especially if mono-periodic. They further split into two sub-groups, one of which follows the same sequence as Miras in the period-luminosity and period-amplitude diagram, without discontinuity. The similarities between Miras and SRVs suggest that the latter can be adopted as complementary distance indicators to the former, thereby at least doubling the available number of LPVs suitable for use as distance indicators. The traditional amplitude-based separation between Miras and SRVs is not necessarily appropriate, and a more physically sound criterion should also involve pulsation periods. While this would require comparatively longer time series, they are expected to become accessible in the coming years even for weak sources thanks to current and future large-scale surveys. The table of reclassified LPVs is made public.
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