**3. Results**

The comparison between the two modern samples highlighted that male individuals show 40% more asymmetry on average. Such enhanced asymmetry is especially evident in the temporoparietal area, the occipital region, and the maxillary bone (Figure 2a). Student's t-test on asymmetry vectors for male and female mean shapes indicates these differences are significant (t = −9.7703, *p* values < 0.001). The asymmetric component is 2.7% of total shape variance. This component is primarily made up of FA, which accounts for 93.6% of it, meaning FA represents 2.53% (93.6 times 2.7) of the total shape variance.

**Figure 2.** Visualization of the degree of asymmetry obtained with *show.asymmetry*. (**a**) Comparison between the mean shape of female and male modern humans (upper row and bottom row, respectively). The range of asymmetry goes from the minimum to the maximum value of asymmetry between the two samples. (**b**) Comparison between the mean shape of female and male modern humans (upper row left and right, respectively) and Petralona and Saccopastore 1 (lower row, left and right respectively). The range of asymmetry is scaled between samples.

When modern humans are compared to Petralona and Saccopastore 1, their degree of asymmetry appears diminutive (Figure 2b). Petralona shows a marked pattern of asymmetry in the temporoparietal area, while the facial complex appears to be more symmetric than the cranial vault. Saccopastore 1 presents a directional pattern of asymmetry with a peak corresponding to the zygomatic and lateral maxillary areas, due to the bad status of preservation of the left side of the splanchnocranium. Both patterns agree well with what has been reported in the literature regarding these specimens [29,37].

Overall, Steinheim is the most asymmetric specimen (Figures 3 and 4). As expected by the descriptions provided in [44], the skull shows extensive deformations on the splanchnocranium, whereas shape was less affected in its rearmost part. However, in keeping with our hypotheses, after the retrodeformation process, the level of asymmetry is close to zero. Lastly, the modern human crania show a minor level of asymmetry when contrasted with the asymmetry level measured in fossil specimens. (Figures 3 and 4).

**Figure 3.** Graphical results produced by *show.asymmetry*. The first column shows the asymmetry pattern in terms of vertex distances between the two superimposed halves (the shared scale is shown on the left). The second column shows the superimposition of the two halves (in this case, the left side is the cyan surfaces, while black wireframe corresponds to the superimposed right side). The third column represents the local area differences between the left and the right side; the scale goes from blue to red (meaning expansion and contraction, respectively), and it is not shared. From top to bottom: mean shape of female individuals, mean shape of male individuals, Petralona, Saccopastore 1, Steinheim before retrodeformation, Steinheim after retrodeformation.

**Figure 4.** Boxplot of the comparison of asymmetry values for the analyzed samples. From left to right: *Homo sapiens* female mean shape, *Homo sapiens* male mean shape, Petralona, Saccopastore 1, and Steinheim (before and after the retrodeformation procedure). The color gradient for the asymmetry displayed in the crania is shown.

The time requested to run all the four case studies was 3.95 s and in particular: Petralona 0.93 s, Saccopastore 0.81 s, Homo sapiens 1.40 s, Steinheim 0.81 s. Speed tests were run with a laptop Intel Core-i7 10875H (2.30 GHz and 32 GB RAM).
