Dear TG,
First of all, I want to thank everyone who has posted on this thread - I've learned things from you ALL that I've not thought of.
It was, TG, St Seraphim of Sarov and his followers, St Seraphim Vyritsky, St Seraphim Zvezdinsky, St Alexandra of Diveyevo et al., that inspired in me an appreciation for the Rosary?Psalter or Rule of the Mother of God from an Eastern point of view.
As a matter of fact, I'm writing an Akathist to Our Lady of the Rosary right now

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From the Eastern point of view, the Western devotion to the Sacred Heart of Jesus presents a number of liturgical/theological issues of a problematic nature.
The display of the organ of Christ's Heart as an object of worship when liturgically we focus on the Whole Christ, as an example.
The Latin Church has also struggled with this point and there is even a directive from Rome NOT to display the Sacred Heart apart from the whole Person of Christ.
There is the issue of invoking the Heart of Christ AS IF it had a personality of its own - although devotees of the Sacred Heart know it is the Lord Jesus they are invoking.
But contemporary Latin devotionals tend to direct prayer to the Person of the Lord Jesus and then make mention of His great love for us in His Heart.
There is the issue, that I don't pretend to completely understand, of why God needed a Human Heart to love us with - Nestorian overtones and the like. I've heard that from Eastern Catholic priests and I will just make mention of it here, but I don't really understand it.
The question then is, as you've inspired just now in your post, TG, is there an Eastern corollary to the Western devotion to the Sacred Heart of Jesus?
And yes there most certainly is.
This is made mention of in St Nicholas Cabasilas' work on the Divine Liturgy, published by St Vladimir's Seminary Press.
Nicholas, who was praised by RC theologians as a "solid Greek theologian," made specific reference to Orthodox worship of the Heart of Christ.
The Orthodox commentary in that book on his words says that Eastern devotion to the Heart of Christ is more liturgical and ecclesial than the "individualistic, pietistic" devotion to the Sacred Heart of Jesus in the West.
I'm just mentioning this to try and understand what is being said here.
A qualitative difference is being drawn in terms of how East vs West liturgically worship Christ under the aspect of His love for humankind and under the aspect of His Heart.
But both traditions clearly DO make use of the symbolism of the Heart.
AND, to make matters even more confusing

, St Nicodemus of the Holy Mountain CONTRADICTS St Alphonsus Liguori when the latter said that the Heart is ONLY a symbol of Christ's Love. Nicodemus says, "No, the Heat is more than merely symbolic - it is where we think and where love is truly generated."
So, I'm going to assume that both traditions, East and West DO worship the Heart of Christ.
But that they do so in different ways.
The East bases its worship entirely on its liturgical tradition where there are MANY references to the Wound in Christ's Side, that we drink and are nourished at the Wound in Christ's Side etc.
The East also has the practice of kissing the base of the Communion of Chalice following Communion as a symbolic act of reverence for the Wound in Christ's Side.
In addition to iconographic Crucifixes depicting the Wounded Side of Christ, from which flows Water and Blood, forgiveness and mercy, there are some icons, like that in my avatar, that depict Christ pointing to the Wound in His Side - Fr. Walter Kern of the Buffalo Archdiocese did an interesting analysis of this.
Other Icons of Christ the Merciful, the Love of Mankind depict the Saviour holding a book that is either decorated with hearts (please see my avatar again) or else holding an open Gospel book with His own words of love written on it such as "I give you another commandment, that you love one another, even as I have loved you" or else "Come to me all you who are weary . . ."
So it is truly the case that devotion to Christ the Lover of Mankind is totally and integrally linked to the liturgical tradition of the Eastern Church - it is really the sum of the Christian Gospel and union with the Lover of Mankind is the sum of the Christian life.
There is also Orthodox devotion to the icon of the "Mother of God of the Seven Swords" or as it is also called "Simeon's Prophecy" (concerning the sword that would pierce her soul or heart).
The official Orthodox Akathist to this icon, which I read in Slavonic last night, is not only incredibly beautiful and inspiring, it does make mention of the heart of the Mother of God, and, at one point, invokes it directly, as one would find in a Latin litany!
Anyway, some food for thought.
Alex