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Axion haloscopes search for dark matter axions from the galactic halo, most commonly by measuring a power excess sourced by the axion effective current density. Constraining axion parameters from detection or lack thereof requires estimating the expected signal power. Often, this is done by studying the response of the haloscope to a known, but different, source current density, for example via a reflection measurement. However, only in the special case when both sources induce the same electromagnetic fields, do the quantities derived from a reflection measurement adequately describe the setup during an axion measurement. While this might be valid for the traditional resonant cavity haloscope, new broadband or open designs like dish antennas or dielectric haloscopes cannot make this assumption. A more general relation between axion- and reflection-induced fields is needed. In this article, we use the Lorentz reciprocity theorem to derive an expression for the axion signal power which instead of the unmeasurable axion-induced fields depends on the measurable reflection-induced fields. This entirely circumvents the need to know the response of the haloscope to the unknown axion source. It applies to a wide variety of haloscopes including resonant cavities, dielectric haloscopes, and broadband dish antennas.