What cells in the body respond to glucagon by breaking down glycogen and releasing glucose?

Liver cells.

Explanation:

Liver maintains the carbohydrate metabolism of the body. Glucose can be stored in the liver in form of glycogen.

Pancreas releases the hormone insulin and glucagon. When the blood glucose levels are high, the insulin hormone is released by pancreas and liver cells convert glucose into glycogen. When the blood glucose level are low, the pancreas releases the hormone glucagon. The liver cell under the hormone influence, breaks down the glycogen and releases glucose. This maintains the blood glucose level in the body.