Conventionally, the I-h symmetry of fullerene C-60 is accepted, which is supported by numerous calculations. However, this conclusion results from the consideration of the molecule electron system, of its odd electrons in particular, in a closed-shell approximation without taking the electron spin into account. Passing to the open-shell approximation has led to both the energy and the symmetry lowering up to C-i. Seemingly contradicting to a high-symmetry pattern of experimental recording, particularly concerning the molecule electronic spectra, the finding is considered in this Article from the continuous symmetry viewpoint. Exploiting continuous symmetry measure and continuous symmetry level approaches, it was shown that formal C-i symmetry of the molecule is by 99.99%I-h. A similar continuous symmetry analysis of the fullerene monoderivatives gives a reasonable explanation of a large variety of their optical spectra patterns within the framework Of the same C-1 formal symmetry exhibiting a strong stability of the C60 skeleton. TOC color pictures present chemical portrait of C-60 in terms of atomic chemical susceptibility (Sheka, Fullerenes: Nanochemistry, Nanomagnetism, Nanomedicine, Nanophotonics; CRC Press: Taylor and Francis Group, Boca Raton, 2011).