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We discuss twisted covariance over the noncommutative spacetime algebragenerated by the relations [q_theta^mu,q_theta^nu]=i theta^{mu nu}, where thematrix theta is treated as fixed (not a tensor), and we refrain from using theasymptotic Moyal expansion of the twists. We show that the tensor nature oftheta is only hidden in the formalism: in particular if theta fulfils the DFRconditions, the twisted Lorentz covariant model of the flat quantum spacetimemay be equivalently described in terms of the DFR model, if we agree to discarda huge non invariant set of localisation states; it is only this last stepwhich, if taken as a basic assumption, severely breaks the relativityprinciple. We also will show that the above mentioned, relativity breaking, adhoc rejection of localisation states is an independent, unnecessary assumption,as far as some popular approaches to quantum field theory on the quantumMinkowski spacetime are concerned. The above should raise some concerns aboutspeculations on possible observable consequences of arbitrary choices of thetain arbitrarily selected privileged frames.

Twisted Covariance as a Non Invariant Restriction of the Fully Covariant DFR Model