We present three-dimensional spectroscopy of 11 E+A galaxies at z= 0.06–0.12. These galaxies were selected for their strong Hδ absorption but weak (or non-existent) [O ii] λ3727 and H emission. This selection suggests that a recent burst of star formation was triggered but subsequently abruptly ended. We probe the spatial and spectral properties of both the young (≲1 Gyr) and old (≳few Gyr) stellar populations. Using the Hδ equivalent widths we estimate that the burst masses must have been at least 10 per cent by mass (Mburst≳ 1010 M⊙), which is also consistent with the star formation history inferred from the broad-band spectral energy distributions. On average the A stars cover ∼33 per cent of the galaxy image, extending over 2–15 kpc2, indicating that the characteristic E+A signature is a property of the galaxy as a whole and not due to a heterogeneous mixture of populations. In approximately half of the sample, we find that the A stars, nebular emission and continuum emission are not co-located, suggesting that the newest stars are forming in a different place than those that formed ≲1 Gyr ago, and that recent star formation has occurred in regions distinct from the oldest stellar populations. At least 10 of the galaxies (91 per cent) have dynamics that class them as ‘fast rotators’ with magnitudes, v/σ, λR and bulge-to-total (B/T) ratio comparable to local, representative ellipticals and S0s. We also find a correlation between the spatial extent of the A stars and the dynamical state of the galaxy such that the fastest rotators tend to have the most compact A star populations, providing new constraints on models that aim to explain the transformation of later type galaxies into early types. Finally, we show that there are no obvious differences between the line extents and kinematics of E+A galaxies detected in the radio (active galactic nucleus, AGN) compared to non-radio sources, suggesting that AGN feedback does not play a dramatic role in defining their properties, and/or that its effects are short.