A simple digraph is semicomplete if for any two of its vertices u and v, at least one of the arcs (u,v) and (v,u) is present. We study the complexity of computing two layout parameters of semicomplete digraphs: cutwidth and optimal linear arrangement (Ola). We prove the following: (1) Both parameters are NP-hard to compute and the known exact and parameterized algorithms for them have essentially optimal running times, assuming the Exponential Time Hypothesis; (2) The cutwidth parameter admits a quadratic Turing kernel, whereas it does not admit any polynomial kernel unless NP ⊆ coNP/poly. By contrast, Ola admits a linear kernel. These results essentially complete the complexity analysis of computing cutwidth and Ola on semicomplete digraphs (with respect to standard parameters). Our techniques also can be used to analyze the sizes of minimal obstructions for having a small cutwidth under the induced subdigraph relation.