This paper presents, to the best of our knowledge, the first study of two-photon absorption (2PA) spectra of a number of symmetrically substituted phthalocyanines in the excitation wavelength region from lambda(ex)=800 to 1600 nm. The selected molecules vary by position of substitution (alpha or beta), number of substituent groups (4, 8, or 16), and presence or absence of metal (Zn or Al) in the center. For all phthalocyanines we find a moderately strong (sigma(2) approximately 100-200 GM), pure electronic, gerade-gerade (g-g) 2PA transition, which shows up as a well-resolved relatively narrow peak in the energy region between Q and B bands (lambda(ex)=870-1100 nm). In metallophthalocyanines (MPcs) this lowest g-g transition is followed by the onset of other higher-frequency 2PA transitions. In some metal-free phthalocyanines (H(2)Pcs) we also reveal a second, broader 2PA transition at slightly higher frequency. In both MPcs and H(2)Pcs, we find a strong monotonic increase of integrated strength of the lowest g-g transition as a function of electron-accepting ability of peripheral substituents, expressed as their aggregated Hammett constant. By using few essential states models (three states for MPcs and four states for H(2)Pcs) we demonstrate the primary role of excited-state transition dipole moment in this effect.