Here we report the first detection of polarization signals induced by the Zeeman effect in spectral lines of the E4Π-A4Π system of FeH located around 1.6 μm. Motivated by the tentative detection of this band in the intensity spectrum of late-type dwarfs, we have investigated the full Stokes sunspot spectrum and have found circular and linear polarization signatures that we associate with the FeH lines of the E4Π-A4Π band system. We investigate the Zeeman effect in these molecular transitions and point out that in Hund's case (a) coupling, the effective Landé factors are never negative. For this reason, the fact that our spectropolarimetric observations indicate that the Landé factors of pairs of FeH lines have opposite signs prompts us to conclude that the E4Π-A4Π system must be in intermediate angular momentum coupling between Hund's cases (a) and (b). We emphasize that theoretical and/or laboratory investigations of this molecular system are urgently needed for exploiting its promising diagnostic capabilities.