The adjective mutual is often used to express that both parties should be in agreement about a yet unknown matter. For example:
The milestone deliveries and criteria of the acceptance tests of the Work shall be determined by the parties in mutual agreement.
Mutual emphasizes that where an agreement requires that something needs to happen, a party is not required to simply consent with what the other party determines (even though such required consent follows a pattern of ‘proposal’, ‘modified proposal’ to ‘approval’). Because ‘agreement’ anyhow implies mutuality, the adjective mutual conveys no more than a friendly worded mood of cooperativeness (where a need to agree might convey a necessity of (tough) negotiation). Apart from this psychological meaning, mutual is redundant.
Another appearance of mutual is in a provision that an “amendment of the Agreement requires the mutual consent of the parties”. Like in agreement, consent is the reciprocal and subjective mindset of two (or more) parties who achieved consensus. Also here, mutual is a pleonasm.
There is one exception, however. American parties consider a statement of approval to be a ‘consent’, as if a consent is a certificate. In such unilateral meaning of consent, adding mutual is understandable. There, it would be more efficient to provide that an “amendment requires the written agreement of the parties” instead of the wordy “a mutual exchange of written consents”.