The Dragon, the Phoenix and the Vortex

1-Small Medallion white ground carpet, 17th/18th c., The Metropoliotan Museum New York

This beautiful and rare white ground Anatolian Small Medallion carpet offers the opportunity of digressing on  ruggy terminology and designs. At issue here is the spandrel decoration: two contrasting color swirling leaves usually called 'Dragon and Phoenix' pattern.
The question arises as to whether terminology can properly convey the source/content of the design. Actually, in fact, lots of fancies are used in the rugdom. That does not mean that they are 'wrong', but they can convey slippery information and some more communication may help. Anyway,  the truth is yet to come.

There does exist in Anatolian carpets a pattern depicting a pairing dragon and a phoenix. Likely swirling in a sort of heavenly chase, the source plausibly originated in Chinese mythology many centuries before it appeared in known rugs.
The earliest survived Chinese carpets dating back to the Ming period don't offer this image, usually depicting the dragon either alone or repeated amidst other celestial and royal signs, i. e. unfolding lotus stems or heavenly clouds. The theme was part of a royal decorative language (pl.3).
Ca va sans dire, both fantastic beings, the dragon and the phoenix, have a far ancient origin and a quite rich symbology interspersed with various cultural milieu cultivated along the Silk Roads at least from the Han period (206BC-220AD), to be conservative (Spirits in Transcultural Skies, Gutschow-Weiler 2015).

2-Dragon palace carpet, Peking (?), 16th  II half, published Der Glanz der Himmelsshoene, Franses -Koenig 2005

3-Dragon and Phoenix, Imperial wucai porcelain box, Wan Li period (1573-1620), published Tesori d'Arte in Cina, Treager-Vainker 2000

Diverse were the appropriations of Chinese imagery in different cultures. Impregnated with a fascination, some designs received new accents and functions migrating out of the Celestial Empire (The Dragon in Transcultural Skies, Its Celestial Aspect in the Medieval Islamic World, Kuhen 2014)).

As to carpets, the first known depiction of the heavenly chasing couple is so far the famous Anatolian rug held in the Museum of Islamic Art in Berlin, dated by the museum to the 15th/16th century (the rug is one of the many finds of Herr W. von Bode). Probably, it is an heir of a class of carpets representing animals dated to the Seljuk and Ilkhanid periods.

4-Dragon and Phoenix carpet, fragment, Anatolia, 15th/16th c., Museum of Islamic art Berlin, photo credits rugrabbit

Whereas animals were an ancient subject in near eastern and eastern textiles since much earlier than the pearl roundel fabrics of the Sassanian and post-Sassanian time, what characterizes this specific image is the composition of the two beings. Probably, a proper term for it is a heavenly dance/chase played by two primal forces represented via fantastic animals: a harmonious interplay governing the cosmos, an idea quite typical of Chinese culture.
But, well before Chinese mythology and cosmology, it is the neolithic art that offers a quite interesting and perhaps influential design: the vortex or spiral. Not yet satisfactorily understood, it may be interpreted as a symbol of the universe, the perennial revolving conceit widespread throughout the world since prehistoric times

 5a-Spiral relief carving, Neolithic Malta, from the Tarxien Temples, 3600-3200 BC.

Similar designs appear in ancient China as well.

5b-Storage Jar, Majiayao culture, Banshan phase, 2650–2300 BCE. Earthenware. Peabody Museum of Archaeology and Ethnology, Harvard University

6-Fantastic Beings and Spiralling Motifs, inlaid bronze mirror, Zhou dynasty, Warring States period (453-221 BC)

7-Unfolding 'arabesque' and Animals, gauze fabric, Zhou dynasty, Warring States period 

And, they appear in Islamic art, where spiraling stems started to surface from the 9th/10th century and soon became part of calligraphy. From the 11th-12th century, the pattern became an autonomous decorative motif.

8-Beyhekim mosque, Konya, Anatolia, mihrab detail with spiraling stem spandrels and embellished script, 13th c. III quarter.

Differently from Persian art, the Ottoman often applies to this pattern the 'orthodoxy' of underlying geometry as seen in the floral designs devised by Master Sinan. That style was to imbue much of the urban artistic output throughout Anatolia and percolated thence in local traditions.
A distinctive lyricism was conveyed by the luxurious rendition of flowers and leaves.

9-Iznik panel, mid 16th, The Louvre

It does not wonder the circle and rotation to be part of these patterns by means of stems and curved leaves, either in the saf or the split-leaf/rumi style. The paired swirling leaves is a frequent appearance in such vocabulary, looking like whirling in a real vortex.

10-leaf and palmette vortex, Karatay Madrasa, Konya, Anatolia, 16th

Obviously enough, this is reflected in carpet designs as best represented by the Ottoman Cairene outcome. Revolving cosmos and unfolding sacredness (the hidden metaphor therein) are common thoughts since prehistoric times and permeate all of the Eurasian religions, Islam included.

11-Cairene Ottoman prayer rug, late 16th, The Walters Art Museum Baltimore

Accordingly to common wisdom, a whole group of Anatolian carpets sourced from the courtly Ottoman tradition, the so-called 'Transylvanian', a specific subgroup of which displays that paired whirling leaves (Antique Ottoman Rugs in Transylvania, Ionescu 2006)

12-'Transylvanian' six columns carpet, 17th II half, Brukhental National Museum 

The motif gracing the upper panel in carpet plate 12 is the same as the one in the spandrel of the white ground carpet at issue. Specifically, the encircled cog-wheel likely replaces a floral device in this particular mix of urban and village decorative tradition.

The main difference with respect to the classical saf tile decoration seen above is in the design of the leaf, which is in the split-leaf/rumi style. A tile in plate 13 exactly shows this type with identical outlining indentations and ornament of the thin end.

13-Iznik tile, 16th century, rotating pattern with split leaf/rumi leaves

More often than not the leaf is befriended by diverse floral forms emerging from the back or hiding it like an awesome early panel displays. And, as the term saz says, that is enchanted forest,  fantastic animals gambol in the midst (The Enchanted Forest in Iznik, Ziffer 2000).

14-Ceramic panel, Sunnet Odasi room, Shah kulu ?, circa 1535, Top Kapi Saray Museum.

The luxurious blossoming forest of the new Ottoman courtly style declares proximity to the ever-inspiring Chinese art, still a model of unrivaled artistry in all of the Islamic lands. The palmette/lotus scroll was, in fact, called Qatha'i, the Turkish name for China  (History of Ornament. From Global to Local, G. Necipoglu 2016) and mythological beasts from that far tradition appeared - a Chinese qilin in the above panel, bottom left. Not strange if also the dragon appears, specifically in some painted folios (pl. 15).  (A whole class of carpets takes its name from the impressive presence of dragons in the decoration, the Caucasian Dragon carpets)

15-Two dragons entwined on a spray of saz foliage, Ottoman painted folio, 1575, LACMA

The name 'Dragon and Phoenix' for the spandrel pattern of the white ground Anatolian opening this entry doesn't even match the dragon and leaves twisted together as seen in typical Ottoman painted folios (fig. 15).  The design in fact does not sufficiently validate the idea of the leaf substituting/representing the phoenix.

Yet finally, the vortex of the two 'chasing' split leaves to an informed and sensitive eye recalls ancient spiraling motifs and the Chinese harmonious reciprocity of the cosmos dance as percolated into western cultures.

16-Dragon and Phoenix, Imperial wucai porcelain box, Wan Li period (1573-1620)

17-Dragon and phoenix, Safavid kilim, Keshan, 16th II half.

Finally, a  splendid 17th  century Azerbaijan embroidery presents our magic whirls interspersed with classic eight pointed medallions. Extremely expressive and spirited, it persuasively conveys the spell this motif had on the contemporaneous eye.

Azerbaijan embroidery, 17th, Sothebys 1995