Comment: 19 pages, 17 figures. Accepted for publication in Journal of Economic Behavior and OrganizationThis paper is concerned with upstreamness and downstreamness of industries and countries. Upstreamness and downstreamness measure respectively the average distance of an industrial sector from final consumption and from primary inputs. Recently, Antr\`as and Chor reported a puzzling and counter-intuitive finding in data from the period 1995-2011, namely that (at country level) upstreamness appears to be positively correlated with downstreamness, with a correlation slope close to $+1$. We first analyze a simple model of random Input/Output tables, and we show that, under minimal and realistic structural assumptions, there is a natural positive correlation emerging between upstreamness and downstreamness of the same industrial sector/country, with correlation slope equal to $+1$. This effect is robust against changes in the randomness of the entries of the I/O table and different aggregation protocols. Secondly, we perform experiments by randomly reshuffling the entries of the empirical I/O table where these puzzling correlations are detected, in such a way that the global structural constraints are preserved. Again, we find that the upstreamness and downstreamness of the same industrial sector/country are positively correlated with slope close to $+1$. Our results strongly suggest that (i) extra care is needed when interpreting these measures as simple representations of each sector's positioning along the value chain, as the ``curse of the input-output identities'' and labor effects effectively force the value chain to acquire additional links from primary factors of production, and (ii) the empirically observed puzzling correlation may rather be a necessary consequence of the few structural constraints (positive entries, and sub-stochasticity) that Input/Output tables and their surrogates must meet.