Deligne cohomology has a geometric interpretation. For example, $H^{2}_{\mathcal{D}}(X,\mathbb{Z}(1))$ is identified with the group $H^{1}(X,\mathcal{O}_{X}^{\ast})$ of isomorphism classes of line bundles on $X$, and $H^{2}_{\mathcal{D}}(X,\mathbb{Z}(2))$ is identified with the group $\mathbb{H}^{1}(X,\mathcal{O}_{X}^{\ast}\xrightarrow{d\log}\Omega^{1}_{X})$ of isomorphism classes of line bundles on $X$ with connection. For the general case see [Gaj97] where the description is in terms of iterated classifying spaces.

Syntomic cohomology is a $p$-adic analogue of Deligne cohomology. Does it have a geometric interpretation along similar lines?

[Gaj97] P. Gajer, Geometry of Deligne cohomology, Invent. Math. 127 (1997), no.1, 155-207.