1. Preheat oven to 325°. Beat butter and shortening at medium speed with a heavy-duty electric stand mixer until creamy. Gradually add sugar, beating until light and fluffy. Add honey, beating until blended. Add eggs, 1 at a time, beating just until blended after each addition.
2. Stir together flour and next 2 ingredients. Add to butter mixture alternately with milk, beginning and ending with flour mixture. Beat at low speed just until blended after each addition. Transfer 2 1/2 cups batter to a 2-qt. bowl, and stir in matcha until blended.
3. Drop 2 scoops of plain batter into a greased and floured 10-inch (16-cup) Bundt pan, using a small cookie scoop (about 1 1/2 inches); top with 1 scoop of matcha batter. Repeat procedure around entire pan, covering bottom completely. Continue layering batters in pan as directed until all batter is used.
4. Bake at 325° for 1 hour and 5 minutes to 1 hour and 15 minutes or until a long wooden pick inserted in center comes out clean.
5. During last 10 minutes of baking, prepare Honeysuckle Glaze. Remove cake from oven, and gradually spoon 1 cup hot Honeysuckle Glaze over cake in pan, allowing glaze to soak into cake after each addition. Reserve remaining glaze. Cool cake completely in pan on a wire rack (about 1 hour and 30 minutes).
6. Remove cake from pan; spoon reserved glaze over cake.