Tangential discontinuities known as cold fronts (CFs) are abundant in groups and clusters of galaxies (GCs). The relaxed, spiral-type CFs were initially thought to be isobaric, but a significant, 10%-20% jump in the thermal pressure P t was reported when deprojected CFs were stacked, interpreted as missing P t below the CFs (i.e. at smaller radii r) due to a locally-enhanced nonthermal pressure P nt . We report a significant (∼ 4.3σ) deprojected jump in P t across a single sharp CF in the Centaurus cluster. Additional seven CFs are deprojected in the GCs A2029, A2142, A2204, and Centaurus, all found to be consistent (stacked: ∼ 1.9σ) with similar pressure jumps. Combining our sample with high quality deprojected CFs from the literature indicates pressure jumps at significance levels ranging between 2.7σ and 5.0σ, depending on assumptions. Our nominal results are consistent with P nt ≃ (0.1-0.3)P t just below the CF. We test different deprojection and analysis methods to confirm that our results are robust, and show that without careful deprojection, an opposite pressure trend may incorrectly be inferred. Analysing all available deprojected data, we also find: (i) a nearly constant CF contrast q of density and temperature within each GC, monotonically increasing with the GC mass M 200 as q ∝ M 0.23±0.04 200 ; (ii) hydrostatic mass discontinuities indicating fast bulk tangential flows below all deprojected CFs, with a mean Mach number ∼ 0.76; and (iii) the newly deprojected CFs are consistent (stacked: ∼ 2.9σ) with a 1.25 +0.09 −0.08 metallicity drop across the CF. These findings suggest that GCs quite generally harbor extended spiral flows.