In the high-transition-temperature (high-T(c)) superconductors the pseudogap phase becomes predominant when the density of doped holes is reduced. Within this phase it has been unclear which electronic symmetries (if any) are broken, what the identity of any associated order parameter might be, and which microscopic electronic degrees of freedom are active. Here we report the determination of a quantitative order parameter representing intra-unit-cell nematicity: the breaking of rotational symmetry by the electronic structure within each CuO(2) unit cell. We analyse spectroscopic-imaging scanning tunnelling microscope images of the intra-unit-cell states in underdoped Bi(2)Sr(2)CaCu(2)O(8 +) (delta) and, using two independent evaluation techniques, find evidence for electronic nematicity of the states close to the pseudogap energy. Moreover, we demonstrate directly that these phenomena arise from electronic differences at the two oxygen sites within each unit cell. If the characteristics of the pseudogap seen here and by other techniques all have the same microscopic origin, this phase involves weak magnetic states at the O sites that break 90 degrees -rotational symmetry within every CuO(2) unit cell.