Pseudospin, an additional degree of freedom emerging in graphene as a direct consequence of its honeycomb atomic structure, is responsible for many of the exceptional electronic properties found in this material. This paper is devoted to providing a clear understanding of how graphene's pseudospin impacts the quasiparticle interferences of monolayer (ML) and bilayer (BL) graphene measured by low-temperature scanning tunneling microscopy and spectroscopy. We have used this technique to map, with very high energy and space resolution, the spatial modulations of the local density of states of ML and BL graphene epitaxially grown on SiC(0001), in presence of native disorder. We perform a Fourier transform analysis of such modulations including wave vectors up to unit vectors of the reciprocal lattice. Our data demonstrate that the quasiparticle interferences associated to some particular scattering processes are suppressed in ML graphene, but not in BL graphene. Most importantly, interferences with 2q F wave vector associated to intravalley backscattering are not measured in ML graphene, even on the images with highest resolution where the graphene honeycomb pattern is clearly resolved. In order to clarify the role of the pseudospin on the quasiparticle interferences, we use a simple model which nicely captures the main features observed in our data. The model unambiguously shows that graphene's pseudospin is responsible for such suppression of quasiparticle interference features in ML graphene, in particular for those with 2q F wave vector. It also confirms scanning tunneling microscopy as a unique technique to probe the pseudospin in graphene samples in real space with nanometer precision. Finally, we show that such observations are robust with energy and obtain with great accuracy the dispersion of the π bands for both ML and BL graphene in the vicinity of the Fermi level, extracting their main tight-binding parameters.