The invagination of cuticle in the alimentary canal of a cockroach is found in both the anterior and posterior parts. This invagination is known as the foregut invagination and hindgut invagination, respectively.

**Foregut Invagination:**
The foregut invagination is located in the anterior part of the alimentary canal of a cockroach. It starts from the mouth and extends up to the proventriculus. The proventriculus is a valve-like structure that separates the foregut from the midgut. The invagination of cuticle in this region forms a tubular structure called the stomodaeum. The stomodaeum is lined with cuticle, which helps in the mechanical breakdown of food particles and the protection of the internal organs. The foregut invagination also contains structures like the salivary glands, the mouthparts, and the esophagus.

**Hindgut Invagination:**
The hindgut invagination is located in the posterior part of the alimentary canal of a cockroach. It starts from the ileum, which is the connection between the midgut and hindgut, and extends up to the anus. The invagination of cuticle in this region forms a tubular structure called the proctodaeum. The proctodaeum is also lined with cuticle, which helps in the absorption of water and the elimination of waste products. The hindgut invagination also contains structures like the Malpighian tubules, which are involved in the excretion of nitrogenous waste.

**Functions of Foregut and Hindgut Invagination:**
1. Mechanical Breakdown: The invagination of cuticle in both the foregut and hindgut helps in the mechanical breakdown of food particles, facilitating digestion.
2. Protection: The cuticle lining in both invaginations protects the internal organs from mechanical damage and provides structural support.
3. Absorption and Excretion: The hindgut invagination is involved in the absorption of water and the elimination of waste products, while the foregut invagination is responsible for the absorption of nutrients.

In conclusion, the invagination of cuticle in the alimentary canal of a cockroach is found in both the anterior (foregut) and posterior (hindgut) parts. This invagination serves various functions, including mechanical breakdown, protection, and absorption/excretion.

In gizzard which is the part of foregut i.e anterior part and also Ileum which is part of hindgut i.e. posterior part of alimentary canal.