You can do this with two systems from different agencies using two different kinds of radio systems on different bands?
Yes, it's possible between different systems, and since it's done switch to switch there is no problem/issue related to frequencies. Typically you would use a 4-wire (E&M) interface for that can be found on many analog and digital systems. (Digital systems typically have it just for the purpose of interconnecting to older systems.)
From what switch to what switch?
Switch = exchange, or possibly base station / repeater to either switch, another base station/repeater or to the dispatch console; whatever is available in the systems you want to interconnect.
How many agencies are actually equipped to do this right from the dispatch console?
I don't know about each and every dispatch consoles but most infra equipment has a 4-wire (E&M) interface somewhere, I have even seen radios (mobiles, not portables) with them.
I've never seen one in my career. Crosspatching usually gets done when the magic comm van from the state shows up three or four hours into the incident.
That's not a technical issue though and perhaps a failure of whoever is in charge to be prepared.
You can do all kind of neat things with TETRA but it only works with other users on the TETRA system.
Sure. I was addressing the issue regarding the inter-agency coms if the radios don't share a common group. If the other system is not a TETRA system then the solution above is available.
As well there are mobile / field-deployable solutions available that link two radios back-to-back, e.g. one TETRA and one analog one, thus allowing interop between different systems / frequencies. Nowadays such equipment is small enough that it can be fitted into any truck, car even, or into a 19" transit case. So when the units roll-out to the scene of the incident they are already equipped or just throw the case into the trunk, and off you go ...
If every agency standardized on one type of radio system in one radio band like 800, interoperability would be a lot easier. That chances of that happening are less than zero.
Partially agree - at least in Europe the problems have been realized and several of the new networks are shared networks with nation-wide coverage. Besides the obvious advantage of easier interops it's also cheaper than every agency owning and maintaining their own network where the OPEX will quickly exceed the CAPEX. The effect is greater the larger the network is, i.e. the CAPEX of regional, state-wide or nation-wide networks is very high but the OPEX will be significantly reduced and result in cost-saving in the long-run.