Hi Micah,
In order to control the label visibility you will need to define map grid edges and toggle the visibility as needed.
Here are the steps:
1. Select the grid from the Contents pane and open its properties (Format Map Grid).
2. From the Options tab, expand the Edges and Corners group and check the Define map grid edges and Preview edges and corners check boxes.
You should see how the edges and corners are defined for you map. Something like this:
3. Click the Components tab.
4. Select Labels.
5. Go to the Visible group and click Selected. You should see indicators reminding you which map edge is which.
6. Select Edge 1. In the example above, edge 1 is the top of the page.
Now it is time to determine which labels are drawn. For edge 1 (top of the page) I want to only display the labels for lines of longitude. That is, lines that go in a North - South direction. Therefore, I want to "turn off" labels for lines of latitude (East - West).
7. Check off West.
This feels a bit counter-intuitive because the cardinal direction in the label does not match the direction for the visibility check box. It is important to keep in mind that the label for visibility check box pertains to the direction of the line and the particular end point of this line.
The unwanted labels are no longer drawing.
8. Repeat this for the other edges.
We designed it this way in ArcGIS Pro to provide as much flexibility as possible to handle not only map rotation cases, but also cases where the map frame is reshaped.
Dan helpfully provided a link above to a help topic that provides more information.
Hope this helps.
Tom