The neutron-deficient mercury isotopes serve as a classical example of shape coexistence, whereby at low energy near-degenerate nuclear states characterized by different shapes appear. The electromagnetic structure of even-mass ^182-188Hg isotopes was studied using safe-energy Coulomb excitation of neutron-deficient mercury beams delivered by the REX-ISOLDE facility at CERN. The population of 0_1,2⁺, 2_1,2⁺ and 4₁⁺ states was observed in all nuclei under study. Reduced E2 matrix elements coupling populated yrast and non-yrast states were extracted, including their relative signs. These are a sensitive probe of shape coexistence and may be used to validate nuclear models. The experimental results are discussed in terms of mixing of two different configurations and are compared with three different model calculations: the Beyond Mean Field model, the Interacting Boson Model with configuration mixing and the General Bohr Hamiltonian. Partial agreement with experiment was observed, hinting to missing ingredients in the theoretical descriptions.