ABSTRACT Large-scale asymmetries (i.e. lopsidedness) are a common feature in the stellar density distribution of nearby disc galaxies both in low- and high-density environments. In this work, we characterize the present-day lopsidedness in a sample of 1435 disc-like galaxies selected from the TNG50 simulation. We find that the percentage of lopsided galaxies (10 per cent–30 per cent) is in good agreement with observations if we use similar radial ranges to the observations. However, the percentage (58 per cent) significantly increases if we extend our measurement to larger radii. We find a mild or lack of correlation between lopsidedness amplitude and environment at z = 0 and a strong correlation between lopsidedness and galaxy morphology regardless of the environment. Present-day galaxies with more extended discs, flatter inner galactic regions, and lower central stellar mass density (i.e. late-type disc galaxies) are typically more lopsided than galaxies with smaller discs, rounder inner galactic regions, and higher central stellar mass density (i.e. early-type disc galaxies). Interestingly, we find that lopsided galaxies have, on average, a very distinct star formation history within the last $10\, \mathrm{Gyr}$, with respect to their symmetric counterparts. Symmetric galaxies have typically assembled at early times (∼8–$6\, \mathrm{Gyr}$ ago) with relatively short and intense bursts of central star formation, while lopsided galaxies have assembled on longer time-scales and with milder initial bursts of star formation, continuing building up their mass until z = 0. Overall, these results indicate that lopsidedness in present-day disc galaxies is connected to the specific evolutionary histories of the galaxies that shaped their distinct internal properties.