I am signing (HMAC) outgoing webhooks to allow users to verify their source, should I also sign outgoing responses?

I am signing (HMAC) outgoing webhooks to allow users to verify their source, should I also sign outgoing responses?

To allow api users to verify the authenticity of outgoing webhooks, I am using a similar model to slack:

• Concatenate timestamp and body, HMAC with pre-shared key, add timestamp and HMAC digest to headers.

• Recipient does the same, and compares to the digest in the header.

I can either implement this exclusively on outgoing webhooks, or I can implement it as middleware that performs this process on both outgoing webhooks, and responses to requests.

Is doing the latter good practice? A good idea?