Aaron,
Regarding the managing the visibility of grid labels, ArcMap does NOT provide any functionality that is not available in ArcGIS Pro.
I had assumed that with different pages in the map series you wanted different labels to display. For example, in created a map book with facing pages you may want the maps for “left side” pages to display the grid labels on the left side of the page, while for “right-side” pages you want the grid labels to display on the right side of the page.
Neither ArcMap, not ArcGIS Pro provide this ability outside the box. You would have to use ArcPy.
Perhaps this is not your workflow. Perhaps by “… orientation of grid labels” you are looking to have only the labels appear for the top and left side of the page. In ArcMap you would toggle the visibility of the Label Axes in the grid properties dialog like so:
Here is an example of 2 map series pages of US states where Hawaii has a map rotation of 0 and California has a map rotation of 90.
This can also be accomplished using ArcGIS Pro. If you are importing an MXD, like the one above, you will likely need to make some adjustments in Pro. For example, importing this MXD results in this for the California page.
Obviously, this is not what it looks like in ArcMap. There is a reason for this. When this MXD is imported into ArcGIS Pro you should get a notification. Click the blue dot to see the notification.
In this case there is a warning that modifications to the grid or graticule may be needed.
In this case ArcGIS Pro is not reading the “Top, Left, Bottom, Right” settings. This concept is different in ArcMap than it is in Pro. With ArcMap, you are restricted to the 4 sides of a page. With Pro, this restriction is removed. For instance, if you reshape a map or map frame so that there are more than 4 edges to the map ArcGIS provides a way to manage the labels for every edge.
For example, here is a reshaped map frame for Hawaii.
There are 8 edges, along with 8 corners. With ArcGIS Pro I can manages the grid labels for each.
You CANNOT do this in ArcMap.
Back to the map imported above, the first step is to define the map grid edges and preview them so we can see the edge numbers we want to work with. For illustration purposes I don’t have labels on in the screenshot.
If it is the top and right edges of the map frame that I want to display labels I need to toggle the visibility for those edges. In this case I want to have the labels on for edges 1 and 1, and off for edges 3 and 4. From the Components tab in the Format Map Grid pane select Labels from the component list and under the Visible group click the Selected button.
Depending on your map rotation values you may not want to toggle all of the “North”, “South”… labels. In the example below edge 1 displays labels for both X and Y (Lat and Long).
In order to longitude labels along edge 1 I would toggle only “North” and “South”. For edge 2 I turned only “East” and “West” only.
For more information on this please see MGRS grids—Layouts | Documentation - and search for "Define map grid edges".
Hope this helps,
Tom