Anharmonicity strongly influences the absorption and emission spectra of polycyclic aromatic hydrocarbon (PAH) molecules. Here, IR-UV ion-dip spectroscopy experiments together with detailed anharmonic computations reveal the presence of fundamental, overtone, as well as 2- and 3-quanta combination band transitions in the far- and mid-infrared absorption spectra of phenylacetylene and its singly deuterated isotopologue. Strong absorption features in the 400-900 cm-1 range originate from CH(D) in-plane and out-of-plane wags and bends, as well as bending motions including the C≡C and CH bonds of the acetylene substituent and the aromatic ring. For phenylacetylene, every absorption feature is assigned either directly or indirectly to a single or multiple vibrational mode(s). The measured spectrum is dense, broad, and structureless in many regions but well characterized by computations. Upon deuteration, large isotopic shifts are observed. At frequencies above 1500 cm-1 for d1-phenylacetylene, a one-to-one match is seen when comparing computations and experiments with all features assigned to combination bands and overtones. The C≡C stretch observed in phenylacetylene is not observed in d1-phenylacetylene due to a computed 40-fold drop in intensity. Overall, a careful treatment of anharmonicity that includes 2- and 3-quanta modes is found to be crucial to understand the rich details of the infrared spectrum of phenylacetylene. Based on these results, it can be expected that such an all-inclusive anharmonic treatment will also be key for unraveling the infrared spectra of PAHs in general.