We show that when observing the range of a chordal SLEκ curve for κ∈(4,8), it is not possible to recover the order in which the points have been visited. We also derive related results about conformal loop ensembles (CLE): (i) The loops in a CLEκ for κ∈(4,8) are not determined by the CLEκ gasket. (ii) The continuum percolation interfaces defined in the fractal carpets of conformal loop ensembles CLEκ for κ∈(8/3,4) (we defined these percolation interfaces in previous work, and showed there that they are SLE16/κ curves) are not determined by the CLEκ carpet that they are defined in.