Previous XMM-Newton observations of the thermally emitting isolated neutron star RX J1605.3+3249 provided a candidate for a shallow periodic signal and evidence of a fast spin down, which suggested a high dipolar magnetic field and an evolution from a magnetar. We obtained a large programme with XMM-Newton to confirm its candidate timing solution, understand the energy-dependent amplitude of the modulation, and investigate the spectral features of the source. We performed extensive high-resolution and broadband periodicity searches in the new observations, using the combined photons of the three EPIC cameras and allowing for moderate changes of pulsed fraction and the optimal energy range for detection. We also investigated the EPIC and RGS spectra of the source with unprecedented statistics and detail. A deep 4σ upper limit of 1.33(6)% for modulations in the relevant frequency range conservatively rules out the candidate period previously reported. Blind searches revealed no other periodic signal above the 1.5% level (3σ; P >  0.15 s; 0.3–1.35 keV) in any of the four new observations. While theoretical models fall short at physically describing the complex energy distribution of the source, best-fit X-ray spectral parameters are obtained for a fully or partially ionized neutron star hydrogen atmosphere model with B = 1013 G, modified by a broad Gaussian absorption line at energy ϵ = 385 ± 10 eV. A double-temperature blackbody model, although a good fit to the source spectrum, is disfavoured as it would require a particularly improbable viewing geometry to explain the lack of X-ray pulsations associated with small hotspots on the surface of the neutron star. We identified a low significance (1σ) temporal trend on the parameters of the source in the analysis of RGS data dating back to 2002, which may be explained by unaccounted calibration issues and spectral model uncertainties. The new dataset also shows no evidence of the previously reported narrow absorption feature at ϵ ∼ 570 eV, whose possible transient nature disfavours an atmospheric origin.Key words: stars: neutron / pulsars: general / X-rays: individuals: RX J1605.3+3249⋆ Based on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA (Target RX J1605.3+3249, large programme 76446; archival data 0073140201, 0073140301, 0073140501, 0157360401, 0671620101).