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Based on the spectral analysis of individual orbits of stars from different $N$-body models, we show that the face-on morphology of the so-called `face-on peanut' bars (boxy bars) and barlenses is supported by different types of orbits. For `face-on peanut' bars, the so-called boxy orbits come to the fore, and they are responsible for the unusual morphology of the bar in the central regions. In the models with compact bulges, the bars show a barlens morphology in their central parts. We found that the barlens supporting orbits come in two types, one of which gives a square-like shape and the other have a rosette-like shape in the frame co-rotating with the bar. Such a shape is typical for orbits around stable loop orbits in nearly axisymmetric potentials only slightly distorted by the bar. They were already known from some of the previous studies but their role in barlens shaping was barely investigated. Although quite simple, the rosette-like orbits are found to be the main building block of a barlens in our models. The detailed analysis of all bar orbits also allowed us to distinguish the x2 orbital family and isolate the structure supported by orbits trapped around this family. The x2 family is well-known, but, apparently, for the first time in $N$-body models we have revealed the structure it supports by means of spectral dynamics and highlight its contribution to the barlens. We found that the x2 family population increases with an increase in central matter concentration.