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<jats:title>ABSTRACT</jats:title> <jats:p>The gravitational lensing equations for convergence, potential, shear, and flexion are simple in polar coordinates and separate under a multipole expansion once the shear and flexion spinors are rotated into a “tangential” basis. We use this to investigate whether the useful monopole aperture–mass shear formulae generalize to all multipoles and to flexions. We re-derive the result of Schneider and Bartelmann that the shear multipole <jats:italic>m</jats:italic> at radius <jats:italic>R</jats:italic> is completely determined by the mass multipole at <jats:italic>R</jats:italic>, plus specific moments <jats:italic>Q</jats:italic> <jats:sup>(<jats:italic>m</jats:italic>)</jats:sup> <jats:sub>in</jats:sub> and <jats:italic>Q</jats:italic> <jats:sup>(<jats:italic>m</jats:italic>)</jats:sup> <jats:sub>out</jats:sub> of the mass multipoles internal and external, respectively, to <jats:italic>R</jats:italic>. The <jats:italic>m</jats:italic> ⩾ 0 multipoles are independent of <jats:italic>Q</jats:italic> <jats:sub>out</jats:sub>. But in contrast to the monopole, the <jats:italic>m</jats:italic> &lt; 0 multipoles are independent of <jats:italic>Q</jats:italic> <jats:sub>in</jats:sub>. These internal and external mass moments can be determined by shear (and/or flexion) data on the complementary portion of the plane, which has practical implications for lens modeling. We find that the ease of <jats:italic>E</jats:italic>/<jats:italic>B</jats:italic> separation in the monopole aperture moments does <jats:italic>not</jats:italic> generalize to <jats:italic>m</jats:italic> ≠ 0: the internal monopole moment is the <jats:italic>only</jats:italic> nonlocal <jats:italic>E</jats:italic>/<jats:italic>B</jats:italic> discriminant available from lensing observations. We have also not found practical <jats:italic>local</jats:italic> <jats:italic>E</jats:italic>/<jats:italic>B</jats:italic> discriminants beyond the monopole, though they could exist. We show also that the use of weak-lensing data to constrain a constant shear term near a strong-lensing system is impractical without strong prior constraints on the neighboring mass distribution.</jats:p>