For the cake: Preheat the oven to 350 degrees F. Mix together the softened butter and 1 tablespoon of the cocoa to make a thick paste. Use this paste to paint the inside of the Bundt pan. Set aside.
In an electric mixer, combine the flour, granulated sugar, baking soda, salt, cinnamon and remaining 1/2 cup cocoa and mix until blended. Add the orange juice, mayonnaise, vinegar and vanilla and mix until just combined, 1 to 2 minutes. Fold in the orange zest. Pour the batter into the prepared pan and bake until a toothpick inserted in the cake comes out clean, 45 minutes. Cool in the pan for 10 minutes, and then turn out onto a cooling rack to cool completely.
For the chocolate glaze: Mix the confectioners' sugar and cocoa together. Whisk in the orange juice, 1 tablespoon at a time, until it reaches a good consistency to drizzle. Drizzle the glaze over the cake.
You can substitute 2/3 cup plain Greek yogurt for the mayonnaise.