We uncover the microscopic mechanism underlying the antiparallel Nd–Fe coupling in the intermetallic 4$f$–3$d$–5$p$ compound Nd${}_3$Fe${}_3$Sb${}_7$ by means of high-field x-ray magnetic circular dichroism (XMCD) spectroscopy, complemented by magnetization measurements and density-functional-theory (DFT) calculations. This coupling is unexpected for intermetallics composed of light rare-earth and transition-metal elements. Sb $M_{4,5}$-edge XMCD measurements indicate a clear spin polarization of Sb 5$p$ states antiparallel to the Fe 3$d$ moments. DFT calculations reveal the site-dependent character of the Sb moments and, together with the Fe–Sb antiparallel alignment seen by XMCD, identify a Fe 3$d$–Sb 5$p$–Nd 5$d$ hybridization pathway for the antiparallel Nd–Fe coupling. Pulsed-field XMCD reveals a field-induced spin reorientation through the evolution of the $c$-axis projections of the Nd and Fe moments. Nd${}_3$Fe${}_3$Sb${}_7$, thus, provides a platform for exploring antiparallel rare-earth–transition-metal ($R$–$T$) coupling mediated by transition-metal 3$d$–ligand $p$–rare-earth 5$d$ hybridization in light-$R$–$T$ intermetallics with $p$-elements.