We study the analytic structure of light-front wave functions (LFWFs) and its consequences for hadron form factors using an explicitly Lorentz-invariant formulation of the front form. The normal to the light front is specified by a general null vector (omega)(sup (mu)). The LFWFs with definite total angular momentum are eigenstates of a kinematic angular momentum operator and satisfy all Lorentz symmetries. They are analytic functions of the invariant mass squared of the constituents M(sub 0)(sup 2) = ((Sigma)k(sup (mu)))(sup 2) and the light-cone momentum fractions x(sub i) = k(sub i)(omega)/p(omega) multiplied by invariants constructed from the spin matrices, polarization vectors, and (omega)(sup (mu)). These properties are illustrated using known nonperturbative eigensolutions of the Wick-Cutkosky model. We analyze the LFWFs introduced by Chung and Coester to describe static and low momentum properties of the nucleons.