Traditional network representations, such as node-link views and adjacency matrices, can show dramatically different visual patterns, depending on the underlying layout or seriation algorithm. In contrast, invariant plots consistently surface the same visual pattern for the same input topology; yet researchers have underexplored them and have not integrated them into visualization systems. We present Syndesmoscope, an interactive system for network exploration that juxtaposes multiple views of the same network. Panes show a familiar a force-directed view alongside three panes with interpretable geometric layouts based on graph-theoretic properties: dense-sparse gradient, geodesic eccentricity, and spectral bisection. As a secondary contribution, we introduce kSnakes, a new invariant plot based on density decomposition. Syndesmoscope supports two key interactions: leapfrogging, or linked highlighting between different and interpretable visual patterns; and hopscotching, or hop-based traversal that extends data selections through the underlying topology. Through usage scenarios across a corpus of 72 diverse networks, we demonstrate how these interactions reveal network patterns inaccessible through any single view alone. Live demo available at https://syndesmoscope.vercel.app/.

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