We have studied the three-phase region associated with the phase inversion of three ternary systems of the type nonionic surfactant (alkylpolyglycolether C(n)E(m)) + water + alkane(B): C8E4 + water + heptane; C8E4 + water + decane; and C4E1 + water + octane. Properties of the coexisting phases (densities, refractive indices, viscosities) and the interfacial tensions were measured in the temperature range from somewhat below the lower critical end-point (T(e)) to somewhat above the upper critical end-point (T(u)) of the three-phase region. It is found that the tensions between the two near-critical phases, sigma-em and sigma-um, respectively, can be represented by the asymptotic critical-point scaling law over almost the entire temperature range of three-phase coexistence. However, whereas a mean-field theory for systems near a tricritical point predicts a symmetrical behaviour of sigma-em and sigma-um in the three-phase region, experiment shows pronounced deviations from symmetry in the two systems with C8E4. The present results are compared with results from the literature on other surfactant systems and systems near a tricritical point. For most systems studied so far, the asymmetry ratio A = sigma(T(u))/sigma(T(e)) has a value A = 1.3 +/- 0.15, regardless of the width of the three-phase region. For the present systems with C8E4, significantly greater asymmetries were found (A = 3.0 for B = heptane and A = 2.0 for B = decane). This deviating behaviour is correlated with the asymmetry in the phase behaviour of these systems, viz. the appearance of a lamellar phase near T(e) but not near T(u).