Matcha, a Japanese green tea powder, adds vivid green color and a delicate flavor. Look for it in Asian grocery stores and gourmet markets, or order online. Store tightly covered in the fridge. (It oxidizes quickly.) If you can't find matcha, pulverize regular green tea in a spice grinder.
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.
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.
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.
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.
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).
Remove cake from pan; spoon reserved glaze over cake.