Learn Hardcore German: Die Teamleiterin bleibt ruhig, obwohl der Termin verschoben wird. - The team leader remains calm, although the appointment is being postponed.

In German, obwohl is a subordinating conjunction (like weil, dass, wenn, etc.).
In clauses introduced by such conjunctions, the conjugated verb goes to the end of the clause.

Main clause: Der Termin wird verschoben. – normal verb-second word order.
Subordinate clause: obwohl der Termin verschoben wird – verb (wird) moves to the end.

So obwohl der Termin verschoben wird literally has the structure:
although the appointment postponed is.

Can I start the sentence with the obwohl-clause?

Yes. You can put the subordinate clause first:

Obwohl der Termin verschoben wird, bleibt die Teamleiterin ruhig.

Two things to note:

You still need a comma between the clauses.
When the subordinate clause comes first, the following main clause still has verb-second word order, so you get:
- bleibt die Teamleiterin ruhig (verb bleibt in second position).

What exactly does obwohl mean here, and how is it different from trotzdem?