While I don't know other general properties (similar to paddability) that imply p-isomorphism, I suppose there is another way that is perhaps more "direct and natural" in the way asked for. Namely, if you can show that two languages are really encoding the same thing. (Indeed, Berman and Hartmanis refer to p-isomorphisms as "polynomial-time recodings" in their introduction, though they don't delve further in this direction.)
Several types of examples:
Example Type 1: Two different string encodings built from "conceptually the same" language. For example, we may encode CNF-SAT instances into $\Sigma^*$ where $\Sigma = \{(,),x,0,1,\wedge,\vee, \neg\}$ in the obvious way, using the bits 0,1 to encode the indices of the variables. Or we might choose any of a number of encodings of CNF-SAT into $\{0,1\}^*$ (for example, using a prefix-free encoding of $\Sigma$). All of these different encodings should yield p-isomorphic languages, where the p-isomorphism is basically to translate from one encoding to the other.
Example Type 2: Using two different data structures to encode the same thing. For example, if you consider the Graph 3-Coloring problem where the input is adjacency matrices, this should be p-isomorphic to one where the input is adjacency lists, incidence matrices, or edge lists. (I guess this is similar to type 1 - if pressed, I suppose I would find it hard to give a technical distinction between "two different encodings of the same language into strings" and "using two different data structures to encode the same objects", but conceptually Type 2 feels a bit more specific than Type 1. Maybe it is a sub-type.)
Example Type 3: Problems that are very easily seen to be equivalent. For example, Clique and Independent Set. If graphs are encoded by adjacency matrices, the p-isomorphism here can literally be given by swapping 0s and 1s. As another example, consider IndSet and VertexCover. Recall a graph has an independent set of size $k$ iff it has a vertex cover of size $|V|-k$. So the isomorphism here is given by mapping $(G,k)$ (an IndSet instance) to $(G,|V|-k)$. As a less trivial possible example, I would be surprised if one could not directly make the holographic equivalence between #$_2$MonRtw3CNF and #$_2$IndSet for cubic graphs from Valiant/Guo-Lu-Valiant into a p-isomorphism, though I have not worked it out carefully.
