We present and discuss new visual wavelength-range observations of the inner regions of the supernova remnant SNR 0540-69.3 that is located in the Large Magellanic Cloud (LMC). These observations provide us with more spatial and spectral information than were previously available for this object. We use these data to create a detailed three-dimensional model of the remnant, assuming linear expansion of the ejecta. With the observations and the model we study the general three-dimensional structure of the remnant, and the influence of an active region in the remnant - a blob - that we address in previous papers. We used the fibre-fed integral-field spectrograph VIMOS at the Very Large Telescope of the European Southern Observatory. The observations provide us with three-dimensional data in [OIII]5007 and [SII]6717,6731 at an 0.33x0.33 spatial sampling and a velocity resolution of about 35 km/s. We decomposed the two, partially overlapping, sulphur lines and used them to calculate electron densities across the remnant at high signal-to-noise ratio. Our analysis reveals a structure that stretches from the position of the blob, and into the plane of the sky at a position angle of about 60 degrees. We speculate that the pulsar is positioned along this activity axis, where it has a velocity along the line of sight of a few hundred km/s. The blob is most likely a region of shock activity, as it is mainly bright in [SII]; future observations of [OII]3729 would be useful to test whether the S/O abundance ratio is higher than average for that location in the remnant. The striking resemblance in X-rays between the pulsar wind nebula (PWN) of SNR 0540-69.3 and the Crab, in combination with our findings in this paper, suggests that the symmetry axis is part of a torus in the PWN. (abridged)
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