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Magnetometry is used to detect ferrous objects at various scales, but detecting small-size, compact sources that produce small-amplitude anomalies in the shallow subsurface remains challenging. Magnetic anomalies are often approximated as dipoles or volumes of dipoles that can be located, and their source parameters (burial depth, magnetization direction, magnetic susceptibility, etc.) are characterized using scalar or vector magnetometers. Both types of magnetometers are affected by space weather and cultural noise sources that map temporal variations into spatial variations across a survey area. Vector magnetometers provide more information about detected bodies at the cost of extreme sensitivity to orientation, which cannot be reliably measured in the field. Magnetic gradiometry addresses the problem of temporal-to-spatial mapping and reduces distant noise sources, but the heading error challenges remain, motivating the need for magnetic gradient tensor (MGT) invariants that are relatively insensitive to rotation. Here, we show that the finite size of magnetic gradiometers compared to the lengthscales of magnetic anomalies due to small buried objects affects the properties of the gradient tensor, including its symmetry and invariants. This renders traditional assumptions of magnetic gradiometry largely inappropriate for detecting and characterizing small-size anomalies. We then show how the properties of the finite-difference MGT and its invariants can be leveraged to map these small sources in the shallow critical zone, such as unexploded ordnance (UXO), landmines, and explosive remnants of war (ERW), using both synthetic and field data obtained with a triaxial magnetic gradiometer (TetraMag).