The Indian genius for elaborate dyeing and painting processes was a particular challenge to equal. The palampore narrative begins in the third gallery with an early-18th-century example made in India but with Japanese styling (a lacquerlike darkness and motif) that was intended to appeal to Dutch tastes of the moment. An especially extravagant example (sixth gallery) is a mid-18th-century Indian palampore embroidered in bright silk on gold cotton. A tree sprouts from a hillock reminiscent of a Chinese scholar’s rock, and its flowers — fantastical to the point of being otherworldly — are so finely stitched that they hardly seem handmade. The palampore saga culminates in the exhibition’s final gallery in a tree-of-life face-off: an Indian dyed version made for Dutch markets, an American appliqué version that resembles an extra-large piece of needlework and a spare, pale, almost neurasthenic Chinese rendition in shimmery silk embroidery on silk satin.

This confrontation is only one of many drop-dead moments in which pictorial power, astonishing skill and cultural cross-germination collude, and where the historical Western infatuation with all things exotic is strikingly apparent. The European absorption of Eastern styles and symbols is especially rich in a late-17th-century English tapestry attributed to the weaver John Vanderbank (sixth gallery), arrayed on a deep blue ground.

Some of its scenes feature turbaned pashas or entwined lovers of a distinctly Indian mien; others are inhabited by Chinese scholars and travelers and even an ornate pagoda. (A bright red parrot is more generally exotic.)

Another tour de force that must have been especially appealing to European aristocrats is a double-sided hanging from 18th-century China with, again, silk embroidery on silk-satin. One side is a deep yellow, the other a soft red. The main motif is a bouquet of flowers whose bold contained forms evince an Indian influence but whose container sits on a fanciful Chinese scholar’s rock pedestal. Miracle of miracles, the embroidery is absolutely identical and exquisitely finished on both sides. Even more memorable is how different the arrangement looks against the different colors, a lesson the Pointillists would surely have loved. It is easy to imagine, as the label suggests, that this was a bed curtain, with the red side visible to the people in bed and the yellow facing outside.

Cultures nearly clash in a towering tapestry of embroidered silk made in China for the Portuguese market in the first half of the 17th century. Dominating the opening gallery, it depicts the abduction of Helen by Paris and his Trojan cohort, based on European prints. Nonetheless, Chinese-style architecture, clouds and waves frame the battling Baroque bodies. The scrum also seems randomly dotted with the heads of Chinese lion-dogs and gorgons that, upon a closer look, appear to be part of the warriors’ gear — perhaps epaulets.

Not all hybridity was a long-distance affair. A handful of textiles in the show are the collaborations of cultures living side by side, as is the case with a wonderful gallery devoted to textiles produced in Spanish-ruled Mexico (where exquisite embroidery was a specialty) and South America (where descendants of the Inca had weaving skills superior to almost any on earth). Here, a fine Peruvian wedding mantle mixes geometric borders of Incan origin with a Spanish lattice pattern, and a Mexican shawl alternates bands of buzzy ikat weave with beguiling, friezelike scenes of people in pleasure boats or at social gatherings. The scenes alternate horizontally with the flowered vines that are among the exhibition’s central motifs.

Intensifying the vividness of the cross-cultural exchange is the sheer physical diversity of the textiles: the shifting techniques and materials tend to sharpen visual perception. There are so many wonders to peruse — and so much to see within each one — that “Interwoven Globe” will benefit as few shows do from repeated visits. Magnifying glasses, which the Met is not supplying, may prove useful.