The first high-resolution Spitzer 1RS 9-37 μm spectra of 29 Seyfert galaxies of the 12 μm Active Galaxy Sample are presented. The spectroscopy was obtained with off-source observations, allowing excellent background subtraction and thus accurate measurements of continuum levels and strengths of weak emission lines. Several new combinations of emission-line ratios and line/continuum ratios were identified as effective diagnostics of the strength of the AGN component in these galaxies. The line ratios [Ne v]/[Ne II], [O IV]/[Ne II], already known, but also [Ne III]/[Ne II] and [Ne v]/[Si II] can be used to measure the dominance of the AGN. The equivalent width of the 11.25 μm PAH feature, like that of the 6.2 μm PAH feature, already studied, also anticorrelates with the dominance of the AGN. Our measurements indicate that the 11.25 μm PAH feature has a constant ratio with H2 S(1) irrespective of Seyfert type, approximately 10 to 1. The ratio of the 19 μm fluxes observed with the two spectrometer channels, with aperture areas differing by a factor of 4, measures the source extendedness, giving another estimate of the AGN dominance, which correlates with both the EWs of [Ne n] and PAH emission. Using the rotational transitions of H2, we estimated temperatures (200-300 K) and masses (1 -10 × 106 M⊙) for the warm molecular component. Finally, we used the ratios of the doublets of [Ne v] and of [S III] to estimate the gas electron density, which appears to be of the order of ne ∼ 103-104 cm-3 for the highly ionized component and a factor 10 lower for the intermediate- ionization gas. © 2008. The American Astronomical Society. All rights reserved.