We report distinctly double-peaked H{alpha} and H{beta} emission lines in the late-time, nebular-phase spectra (>~200 days) of the otherwise normal at early phases (<~100 days) type IIP supernova ASASSN-16at (SN2016X). Such distinctly double-peaked nebular Balmer lines have never been observed for a type II SN. The nebular-phase Balmer emission is driven by the radioactive ^56^Co decay, so the observed line profile bifurcation suggests a strong bipolarity in the ^56^Ni distribution or in the line-forming region of the inner ejecta. The strongly bifurcated blueshifted and redshifted peaks are separated by ~3x10^3^km/s and are roughly symmetrically positioned with respect to the host-galaxy rest frame, implying that the inner ejecta are composed of two almost-detached blobs. The red peak progressively weakens relative to the blue peak, and disappears in the 740 days spectrum. One possible reason for the line-ratio evolution is increasing differential extinction from continuous formation of dust within the envelope, which is also supported by the near-infrared flux excess that develops after ~100 days.