Is there a word like that that can be used to describe such an action, event? I used to work at a software firm where there were a lot of separate teams and they would sometimes work together for a big project and before that we would meet up to know each other.
You could call this event confluence:
a coming together of people or things; concourse.
Confluence means a flowing together. In a literal sense, it's about rivers. But it's more often used to talk about the coming together of factors or ideas, or of cultures in a diverse city.
I work in a company not unlike the one you describe, and meetings where groups get together at the start of a project or a period of joint work are often called "kick-off" meetings, because they start the project, whatever it is. (A sports analogy.) Sometimes they are just called "introductory" or "getting-to-know-you" meetings. Not very eloquent terms, but they are pretty clear.