We study duality between input and output in the π-calculus. In dualisable versions of π, including πI and fusions, duality breaks with the addition of ordinary input/output types. We introduce $\overline\pi$, intuitively the minimal symmetrical conservative extension of π with input/output types. We prove some duality properties for $\overline\pi$ and we study embeddings between $\overline\pi$ and π in both directions. As an example of application of the dualities, we exploit the dualities of $\overline\pi$ and its theory to relate two encodings of call-by-name λ-calculus, by Milner and by van Bakel and Vigliotti, syntactically quite different from each other.