Thanks, Skye. I continued to puzzle over the issue this morning, and I discovered the problem lies in my layout templates. All of our company layout templates default to the "Iterations" tab not being visible when the user is viewing a "Scrum Team". All the other available tabs default to visible, but not the "Iterations" tab.
The controls for Scrum Team, Kanban Team, and Waterfall Team within layout templates is something new with New Experience--so I suppose it's equally likely this is a "bug" or a "feature." (Any feedback from the Workfront folks?)
In any case, I've updated the relevant layout templates so that I and my team can view the Iterations tab again. Something to consider if/when you switch your teams over to the New Experience.
Thanks again!
~Eric