For overflow and more menus, Google Material Design seems to favour vertical dot menus, see
Apple seems to favour horizontal dot menus (see http://imgur.com/BH2NioE).
On desktop applications, horizontal dots mean "this menu action will pop over a dialog box with things for you to fill in". Arrows, not dots, mean "show more menu options".
Horizontal dots on touch devices can mean "swipe horizontally to see the next card/image".
So already the horizontal menu icon is overloaded.
I've read that some designers feel that we should use the dot menu that matches how the menu will pop up. I.e., use vertical dot menu for when the menu pops down or up, and use horizontal menu for when the menu pops left or right.
So, with all that said, is the cognitive load of using both icons in the same experience worth the guidance of how the menu will appear? There's added load in two ways: making the user wonder "what's the difference between these two icons? Do they behave differently?" and to recognize two different icons.
In what way does providing the information of in which direction the menu will pop help the user? Does the user really need to know where it's going to appear?
You can probably tell I'm not sold on the idea of using both menu icons in the same UI and am leaning toward using just the vertical menu icon (when I don't have room for a text menu label), so I'd love to hear some counter-arguments in support of doing so to ensure I'm not overlooking something.