Spatial attention supports navigation by prioritizing information from selected locations. A candidate neural mechanism is provided by theta-paced sweeps in grid- and place-cell population activity, which sample nearby space in a left-right-alternating pattern coordinated by parasubicular direction signals. During exploration, this alternation promotes uniform spatial coverage, but whether sweeps can be flexibly tuned to locations of particular interest remains unclear. Using large-scale Neuropixels recordings in freely-behaving rats, we show that sweeps and direction signals are rapidly and dynamically modulated: they track moving targets during pursuit, precede orienting responses during immobility, and reverse during backward locomotion — without prior spatial learning. Similar modulation occurs during REM sleep. Canonical head-direction signals remain head-aligned. These findings identify sweeps as a flexible, attention-like mechanism for selectively sampling allocentric cognitive maps. ### Competing Interest Statement The authors have declared no competing interest. European Research Council, Synergy Grant 951319 (EIM) The Research Council of Norway, Centre of Neural Computation 223262 (EIM, MBM), Centre for Algorithms in the Cortex 332640 (EIM, MBM), National Infrastructure grant (NORBRAIN, 295721 and 350201) The Kavli Foundation, https://ror.org/00kztt736 Ministry of Science and Education, Norway (EIM, MBM) Faculty of Medicine and Health Sciences; NTNU, Norway (AZV)

Sequence transformations, like visual motion, dominate the world around us, but are poorly handled by current models. We introduce the first flow equivariant models that respect these motion symmetrie...