  In this paper, we revisit the folded node and the bifurcations of secondary
canards at resonances $\mu\in \mathbb N$. In particular, we prove for the first
time that pitchfork bifurcations occur at all even values of $\mu$. Our
approach relies on a time-reversible version of the Melnikov approach in
\cite{wechselberger2002a}, used in \cite{wechselberger_existence_2005} to prove
the transcritical bifurcations for all odd values of $\mu$. It is known that
the secondary canards produced by the transcritical and the pitchfork
bifurcations only reach the Fenichel slow manifolds on one side of each
transcritical bifurcation for all $0<\epsilon\ll 1$. In this paper, we provide
a new geometric explanation for this fact, relying on the symmetry of the
normal form and a separate blowup of the fold lines. We also show that our
approach for evaluating the Melnikov integrals of the folded node -- based upon
local characterization of the invariant manifolds by higher order variational
equations and reducing these to an inhomogeneous Weber equation -- applies to
general, quadratic, time-reversible, unbounded connection problems in $\mathbb
R^3$. We conclude the paper by using our approach to present a new proof of the
bifurcation of periodic orbits from infinity in the Falkner-Skan equation and
the Nos\'e equations.
