Washing of Hands. The Lavabo, an expression of the priest's desire to be cleansed within, has been omitted, probably because it is viewed that there is no longer a practical reason for this rite.
Orate, fraters. This prayer has also been omitted, presumably because of the mention of sacrifice. This may have been done for ecumenical reasons. However, a priest long associated with the NC Way once told me that the sacrificial nature of the Mass was a pre-Vatican II notion and not of particular concern today!
Agnes Dei. Has been omitted.
Domine, non sum dignus ... et sanabitur anima mea. This response has also been omitted from the celebration. I can not comment on why this omission has taken place but the attitude of the catechists seems to be that, no matter how sinful we are, if we have been invite to a meal we should eat. Hence, Spiritual Communion is not encouraged.
As "primitive" as the Novus Ordo is supposed to be, these people take the liturgy back even further, and play-act that they are "pure" "Early Christians."
At my parish, some of them may attend the regularly scheduled public Masses with the rest of the parishoners, but they will not communicate with everyone else. They will save that for Saturday evening, when they go down to the "basement/meeting hall" level of the church building, where they hold their own special Neo-Catechumenal Mass.
The Neo-Catechumenal Mass is not advertised on the plaque outside the church, nor in the bulletin. It is by invitation only! Someone who belongs has to "tap" you and "invite" you to this "special" liturgy.
It is a gnostic cult within the Catholic Church. The people I know are nice and all, and do hard work around the parish, etc. But there is a sort of glint in their eyes, like "we belong to a better, purer Church than you do!"