PSR J1509-5850 is a middle-aged pulsar with a period of P ≈ 89 ms and spin-down power of E_dot=5.1× 1035 erg s-1, at a distance of about 3.8 kpc. We report on deep Chandra X-ray Observatory observations of this pulsar and its pulsar wind nebula (PWN). In addition to the previously detected tail extending up to 7‧ southwest from the pulsar (the southern outflow), the deep images reveal similarly long, faint, diffuse emission stretched toward the north (the northern outflow) and the fine structure of the compact nebula (CN) in the pulsar vicinity. The CN is resolved into two lateral tails and one axial tail pointing southwest (a morphology remarkably similar to that of the Geminga PWN), which supports the assumption that the pulsar moves toward the northeast. The luminosities of the southern and northern outflows are about 1× 1033 and 4× 1032 erg s-1, respectively. The spectra extracted from four regions of the southern outflow do not show any softening with increasing distance from the pulsar. The lack of synchrotron cooling suggests a high flow speed or in situ acceleration of particles. The spectra extracted from two regions of the northern outflow show a hint of softening with distance from the pulsar, which may indicate slower particle propagation. We speculate that the northern outflow is associated with particle leakage from the bow-shock apex into the ISM, while the southern outflow represents the tail of the shocked pulsar wind behind the moving pulsar. We estimate the physical parameters of the observed outflows and compare the J1509-5850 PWN with PWNe of other supersonically moving pulsars. Read more here.