This paper presents a single-pixel polarimetric compressive sensing (CS)-based direction of arrival (DoA) estimation technique using a cavity backed programmable coding metasurface aperture. The single-pixel DoA retrieval technique relies on a dynamically modulated waveform diversity, enabling spatially incoherent radiation masks to encode the incoming plane waves on the radar aperture using a single channel. The polarimetric nature of the wave-chaotic coding metasurface ensures that DOA estimation considers the sensitivity of the structure to the polarization state of the incoming waves. We show that the polarimetric single-pixel DoA concept can be enabled by encoding the polarization information of the incoming waves at the physical layer level within the antenna. A dynamically reconfigurable wave-chaotic metasurface, that possesses a structured sparsity of dual-polarized coded metamaterial elements, is proposed for the proof of concept. It is also shown that by encoding and compressing the source generated far-field incident waves into a single channel using spatio-temporal incoherent measurement modes generated by the coded programmable metasurface, we can retrieve high fidelity polarimetric DoA information from compressed measurements.