Personally, I would like it all alike.

You could build your own fence horizontally next to your neighbor's vertical fence (same height) making all sides, back and front the same. If you have children they will climb on the horizontal fence all the time as well as criminals will have easier access to your yard. I have had both types of fences and the vertical allows for better air flow if boards are alternated on the front and back.

If the front horizontal part of the fence is the only part of the fence (no sides or back) visible when looking at the house from the street I see no reason you can't do the horizontal in front. Who will know except the neighbors? Hopefully the horizontal will work with the house's architecture.

I need to see the fence you plan , the fence that is there and pics of your house and how it relates to the existing fence.