In a 3- to 4-quart pan over high heat, bring 7 1/4 cups water, distilled vinegar, wine vinegar, salt, and pickling spice to a boil. Reduce heat, cover, and simmer, stirring occasionally, for 15 minutes.
Follow steps 1 through 4 of Canning Instructions, using six quart-size jars.
Snap off and discard tough ends from asparagus.
In an 8- to 10-quart pan over high heat, bring 4 quarts water to a boil. Add half the asparagus and cook until tender-crisp to bite, about 2 minutes. With a slotted spoon, transfer to a large bowl of ice water. Repeat to cook remaining asparagus and transfer to ice water. Add half the carrots to pan and cook until tender-crisp to bite, about 3 minutes; transfer to ice water. Repeat to cook remaining carrots and transfer to ice water. Drain vegetables. Trim asparagus to 6-inch spears, reserving ends.
Place 2 cloves garlic and 1 chile in each jar. Into five jars, equally pack asparagus spears and about 3/4 of the carrots vertically without forcing. Pack asparagus trimmings and remaining carrots into remaining jar.
Follow steps 5 through 11 of Canning Instructions, leaving 1/2 inch of headspace when you pour hot vinegar mixture over vegetables in each jar. Bring water in canner to 180°-185° and process for 30 minutes.
Note: Nutritional analysis is per 1/2 cup drained.