It is called a "mitre."
The bishop's headdress is called a mitre.
A mitre
The Bishop headdress is called a mitre.
In the Catholic Church most Bishops head a diocese.
It is a dhari (A dancer's headdress)
double crown
The ceremonial hat of a bishop is called a mitre. It is a tall, pointed hat worn during liturgical ceremonies as a symbol of the bishop's authority and office within the Catholic Church.
When viewed from the South it has an appearance like a Christian Bishop's Mitre or Headdress. Mitre Peak is in Fiordland, New Zealand.
The possessive form of the singular noun headdress is headdress's.example: The headdress's height was as tall as the dancer wearing it.
The singular possessive form of "headdress" is "headdress's."
In Eastern Orthodoxy, the Church Year or Liturgical Year is referred to as "the Indiction."
I think you're talking about a bridle.