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 Visual Multi-Object Spectrograph at the Very Large Telescope of the European Southern Observatory. The observations provide us with three-dimensional data in [O iii] λ5007 and [S ii] λλ6717, 6731 at a 0.33 arcsec × 0.33 arcsec spatial sampling and a velocity resolution of about 35 km s^{− 1}. We decomposed the two, partially overlapping, sulphur lines and used them to calculate electron densities across the remnant at a high signal-to-noise ratio. In our study, we recover results of previous studies, but we are more importantly able to obtain more detailed information than before. 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 PA ≃ 60°. Assuming a remnant age of 1000 yr and the usual LMC distance, the structure has an inclination angle of about 65° to the line of sight. The position angle is close to the symmetry axis with present and past activity in the visual and the X-ray wavelength ranges. 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^{− 1}. The ‘blob’ is most likely a region of shock activity, as it is mainly bright in [S ii]; future observations of [O ii] λλ3726, 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. This is in agreement with the original suggestion by Gotthelf & Wang.