  Applications of tangles of connectivity systems suggest a duality between
these, in which for two sets $X$ and $Y\!$ the elements $x$ of $X$ map to
subsets $Y_x$ of $Y\!$, and the elements $y$ of $Y\!$ map to subsets $X_y$ of
$X$, so that $x\in X_y$ if and only if $y\in Y_x$ for all $x\in X$ and $y\in
Y\!$. We explore this duality, and relate the tangles arising from the dual
systems to each other.
