We present integral field spectroscopic observations of the central region of the active galaxy NGC 4258 obtained with the fibre IFU system INTEGRAL. We have been able to detect cold neutral gas by means of the interstellar NaD doublet absorption and to trace its distribution and kinematics with respect to the underlying disc. The neutral gas is blueshifted with projected velocities in the 120-370kms-1 range. We have also detected peculiar kinematics in part of the ionized gas in this region by means of a careful kinematic decomposition. The bipolar spatial distribution of the broader component is roughly coincident with the morphology of the X-ray diffuse emission. The kinematics of this gas can be explained in terms of expansion at very high (projected) velocities of up to 300 km s-1. The observations also reveal the existence of a strip of neutral gas, parallel to the major kinematic axis, that is nearly coincident with a region of very high [SII]/Hα ratio tracing the shocked gas. Our observations are consistent with the jet model presented by Wilson, Yang & Cecil in which a cocoon originating from the nuclear jet is shocking the gas in the galaxy disc. Alternatively, our observations are also consistent with the bipolar hypershell model of Sofue and Sofue & Vogler. On balance, we prefer the latter model as the most likely explanation for the puzzling features of this peculiar object.