We study instantaneous two-photon absorption (2PA) in a series of nominally quasi-centrosymmetric trans-bis(tributylphosphine)-bis-(4-((9,9-diethyl-7- ethynyl-9H-fluoren-2-yl) ethynyl)-R)-platinum complexes, where 11 different substituents, R = N(phenyl)(2)(NPh2), NH2, OCH3, t-butyl, CH3 H, F, CF3, CN, benzothiazole, and NO2, represent a range of electron-donating (ED) and electron-withdrawing (EW) strengths, while the Pt core acts as a weak ED group. We measure the 2PA cross section in the 540-810 nm excitation wavelength range by complementary femtosecond two-photon excited fluorescence (2PEF) and nonlinear transmission (NLT) methods and compare the obtained values to those of the Pt-core chromophore and the corresponding noncentrosymmetric side group (ligand) chromophores. Peak 2PA cross sections of neutral and ED-substituted Pt complexes occur at S-0 -> S-n transitions to higher energy states, above the lowest-energy So -> S, transition, and the corresponding values increase systematically with increasing ED strength, reaching maximum value, sigma(2) similar to 300 GM (1 GM = 10(-50) cm(4) s), for R = NPh2. At transition energies overlapping with the lowest-energy S-0 -> S-1 transition in the one-photon absorption (IPA) spectrum, the same neutral and ED-substituted Pt complexes show weak 2PA, sigma(2) < 30-100 GM, which is in agreement with the nearly quadrupolar structure of these systems. Surprisingly, EW-substituted Pt complexes display a very different behavior, where the peak 2PA of the S-0 -> S-1, transition gradually increases with increasing EW strength, reaching values sigma(2) = 700 GM for R = NO2, while in the So -> S-n transition region the peak 2PEF cross section decreases. We explained this effect by breaking of inversion symmetry due to conformational distortions associated with low energy barrier for ground-state rotation of the ligands. Our findings are corroborated by theoretical calculations that show large increase of the permanent electric dipole moment change in the S-0 -> S-1 transition when ligands with strong EW substituents are twisted by 90 degrees relative to the planar chromophore. Our NLT results in the S-0 -> S-1, transition region are quantitatively similar to those obtained from the 2PEF measurement. However, at higher transition energy corresponding to S-0 -> S-n transition region, the NLT method yields effective multiphoton absorption stronger than the 2PEF measurement in the same systems. Such enhancement is observed in all Pt complexes as well as in all ligand chromophores studied, and we tentatively attribute this effect to nearly saturated excited-state absorption (ESA), which may occur if 2PA from the ground state is immediately followed by strongly allowed IPA to higher excited states.