I have a polygon layer made up of grid cells, illustrating a survey system: 
![enter image description here][1]


I'm trying to create a point on each corner of the polygon. I've used a geoprocessing tool called "Feature Vertices to Points" in ArcGIS, but what it does is create a point for each vertice:

![enter image description here][2]


  [1]: https://i.sstatic.net/dQ8Uv.png
  [2]: https://i.sstatic.net/NHglY.png

Is there a tool out there that will allow for the creation of corner points? 

Please note this is a small subsection of the survey grid that I clipped out to run a proof of concept. 

What I'm basically trying to do is create these points of this grid (which was created many years ago) and then apply the same methodology to the latest version of the survey grid, which is based off of NAD83, to see how much and where the grids have shifted. 

The differences in the grids are important when determining how measurements change when it comes to locating buildings, wells etc.