We report on six new Chandra observations of the Geminga pulsar wind nebula (PWN). The PWN consists of three distinct elongated structures - two $approx 0.2 d_{250}$ pc long lateral tails and a segmented axial tail of $approx 0.05 d_{250}$ pc length, where $d_{250}=d/(250 {rm pc})$. The photon indices of the power law spectra of the lateral tails, $Gamma approx 1$, are significantly harder than those of the pulsar ($Gamma approx 1.5$) and the axial tail ($Gamma approx 1.6$). There is no significant diffuse X-ray emission between the lateral tails -- the ratio of the X-ray surface brightness between the south tail and this sky area is at least 12. The lateral tails apparently connect directly to the pulsar and show indication of moving footpoints. The axial tail comprises time-variable emission blobs. However, there is no evidence for constant or decelerated outward motion of these blobs. Different physical models are consistent with the observed morphology and spectra of the Geminga PWN. In one scenario, the lateral tails could represent an azimuthally asymmetric shell whose hard emission is caused by the Fermi acceleration mechanism of colliding winds. In another scenario, the lateral tails could be luminous, bent polar outflows, while the blobs in the axial tail could represent a crushed torus. In a resemblance to planetary magnetotails, the blobs of the axial tail might also represent short-lived plasmoids which are formed by magnetic field reconnection in the relativistic plasma of the pulsar wind tail.
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