This paper is devoted to the study of codimension-one reversible Hopf bifurcation; more precisely, the authors study periodic solutions near an equilibrium whose eigenvalues collide on the imaginary axis, where such a collision arises persistently in a one-parameter family. This situation is also known as 1:1 resonance. In reversible equivariant dynamical systems, there are three types of codimension-one bifurcations of eigenvalues near 1:1 resonance: splitting (from the imaginary axis to the complex plane), passing (on the imaginary axis) and crossing (the imaginary axis). In this paper, the author focus on the case, where Hopf bifurcation takes place with a pair of doubly degenerate eigenvalues on the imaginary axis, with all other eigenvalues nonresonant with respect to the doubly degenrate eigenvalues. This is the simplest context in which reversible equivariant Hopf bifurcation may arise, leading via the Lyapunov-Schmidt reduction to a bifurcation problem in \(\mathbb{R}^{4}\).