Q:

Cortico-cortical connections

I wonder what the principal ways of (direct) cortico-cortical connections are.
I came up with the following possibilities:

vertical: a cortical neuron connects to another one in roughly the same cortical column, but in another layer
horizontal: a cortical neuron connects to another nearby one, preferably in the same or neighboring layer
"intragyric": a cortical neuron connects to another one in the same gyrus (see here)
interhemispheric: a cortical neuron connects to another one in the same (or neighboring) area of the opposite hemisphere

Are all of these relevant? And/or are there relevant but completely different ways of cortico-cortical connections, e.g. "intergyric"?

Addendum: Maybe too late I found a concise summary in the Wikipedia articles on white matter and association fibers (the latter term I wasn't aware of when I asked my question), which doesn't contradict and complements Bryan's answer:

intracortical (Bryan: "intracolumnar" vs. "horizontal")
short association fibers ("also called arcuate or U-fibers [...]  connect together adjacent gyri")
long association fibers which "connect the more widely separated gyri and are grouped into bundles" (often called fasciculi, but probably not exclusively)
commissural fibers (Bryan: "callosal projections")

There definitely seems to be no reserved name for something like "intragyric" fibers (see second addendum here).

A:

Your question "are all of these relevant" doesn't make much sense to me, but I am going to assume you are talking about it in terms of terminology and answer that way.
The "vertical" connections you describe are usually referred to in the context of cortical "columns" and are therefore referred to as "intracolumnar." Within a column, intralaminar connections by far dominate the total number of synapses, but connections between layers are also important (note that my answer Cortical projections from layers 2/3 back to 4? covers some of this as well, though that question is much more specific). 
Horizontal connections refer to connections between cortical columns of the same cortical area (e.g., from V1 to V1 with a different orientation selectivity). These are often stronger between the same layers, but simply saying "horizontal" does not specify that you mean intralaminar horizontal connections.
"Intragyric" is not a word that is used - I think you made it up. Gyri aren't really specialized functional units, and although there may be some brain areas that are localized within a gyrus many span either several gyri or share a gyrus with other areas (for example, "Heschl's gyrus" isn't even just one gyrus). Therefore, you can't really generalize "gyric" connections as being in a particular class, you would have to refer to the specific areas that are connected to understand the nature of those projections.
Interhemispheric connections are fairly ubiquitous in neocortex, and they make up several white matter tracts that connect the hemispheres, most prominently the corpus callosum. As such, those connections are also often referred to as "callosal" projections.
Other terminologies that are used are "top-down" versus "bottom-up" according to the position of areas in the cortical hierarchy. "Lower" areas are considered to be those with primary sensory and motor functions, whereas higher areas are associative/executive areas. The canonical reference for these projections is Felleman, D. J., & Van Essen, D. C. (1991). Distributed hierarchical processing in the primate cerebral cortex. Cerebral cortex (New York, NY: 1991), 1(1), 1-47. The key feature of top-down versus bottom-up projections is a difference in the interlaminar pattern of connectivity between different cortical areas. It can be hard to really parse the precise hierarchical of some of the higher-order cortical areas, and the patterns of top-down and bottom-up are fairly mixed for areas at similar levels of the hierarchy, however.

