Carmelite spirituality is not contemplative and apostolic. It is apostolic because it is contemplative.
This is true because the soul is wholly subject to the action of the Holy Spirit. At every level of the spiritual life, the Holy Spirit is at work: purifying, enlightening, unifying, transforming. The Spirit is in the soul like an interior fountain, the source of its union with God. Is he not the Spirit of Love? Guided by the Holy Spirit, the soul acquires great simplicity in its abandonment to the divine action, an awareness of a living, spiritual continuity, as well as a constantly renewed élan.
The Holy Spirit is the source of contemplative life as well as the life of the apostolate. It is the Spirit who draws us into himself for prayer and sends us out to conquer the world.
Paul-Marie of the Cross, O.C.D.
This classic definition of Carmelite spirituality comes from an essay by Father Paul-Marie that appeared in a 1953 survey of Catholic spirituality by Father Jacques Gautier, who was the director of the Sulpician Seminary in Paris. You can read the quote in context here.
Father Paul-Marie of the Cross (Paul Hayaux du Tilly) was born in Paris in 1902. He studied at the Sorbonne and the Paris archdiocesan seminary and was ordained a priest of the archdiocese in 1933. He was the assistant director of the historic private Catholic academy École Gerson in Paris before he discerned his call to Carmel and entered the Paris province of the Discalced Carmelites in 1941, taking the name Paul-Marie of the Cross.
When Father Paul-Marie’s confrère Father Jacques of Jesus was arrested by the Nazis for offering shelter to Jewish students at the Carmelite boarding school—the same Père Jacques made famous in the film Au Revoir les Enfants—Father Paul-Marie was the logical choice to become director of the boarding school, Little College of St. Therese of the Child Jesus.
He later served as a prior in Lille, France and was in charge of spirituality programs teaching Carmelite prayer. His writings are numerous, with translations in multiple languages. The only book that is currently in print is Sacred Heart Sister Kathryn Sullivan’s 1959 translation, Carmelite Spirituality in the Teresian Tradition, which has been revised and edited by ICS Publications, the publishing house of the Discalced Carmelite Friars, Washington Province.
of the Cross, P-M, Payne, S 1997, Carmelite Spirituality in the Teresian Tradition, ICS Publications, Washington DC.