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Recently Kloeckner described the structure of the isometry group of the quadratic Wasserstein space W2(R n ). It turned out that the case of the real line is exceptional in the sense that there exists an exotic isometry flow. Following this line of investigation, we compute Isom (Wp(R)), the isometry group of the Wasserstein space Wp(R) for all p ∈ [1, ∞) \ {2}. We show that W2(R) is also exceptional regarding the parameter p: Wp(R) is isometrically rigid if and only if p = 2. Regarding the underlying space, we prove that the exceptionality of p = 2 disappears if we replace R by the compact interval [0,1]. Surprisingly, in that case, Wp([0, 1]) is isometrically rigid if and only if p = 1. Moreover, W1([0, 1]) admits isometries that split mass, and Isom (W1([0, 1])) cannot be embedded into Isom (W1(R)).

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