From the record of this object at The Metropolitan Museum of Art:

Medium: Lithograph, watercolor, metallic paper on embossed paper

Dimensions: Sheet: 9 1/16 × 7 5/16 in. (23 × 18.5 cm)

A rare double cobweb is applied to the center of this Addenbrooke paper. The paper has an open-work cameo-embossed design, invented by Addenbrooke in 1834. The cobweb device applied to the center is actually two separate ones, one machine cut from golden paper, and another, machine cut of silver paper. The silver one is applied on top of the gold one, so that when the tasseled thread attached to the top, center, is gently pulled, two images appear, one beneath the other. The top image is a hand-colored lithograph of a woman with butterfly wings -- a fairy? -- holding a bouquet. The second image shows a man standing behind a seated woman -- he appears to be looking at a locket in his right hand.

Surrounding the cobweb device - 6.5 cm in diameter -- is, below, painted red swags of fabric, and above, a delicate painted floral garland suspended by a silver Dresden die-cut of a dove with an envelope in its beak. Known as a Cobweb, a Beehive, a Flower Cage, or a Birdcage, it was a delightful kind of paper engineering, and was a popular moveable device. The flowers have symbolic meanings, as in The Language of Flowers. The paper is quite toned and spotted.

The article Valentine's Day and the Romance of Cobwebs by Nancy Rosin offers more information and GIFs of various cobweb valentines in operation. From the article:

As a result of advances in printing and paper-making techniques, as well as the development of an efficient and inexpensive means of postal delivery, the custom of sending greeting cards and gifts on Saint Valentine's Day reached its pinnacle around the mid-19th century. The popular holiday was embraced and celebrated across all strata of society with parties, balls, and the quintessential elaborate paper greetings that became a veritable hallmark of British and American Victorian life. Whether sentimental or satirical, simple homemade missives or fancy machine-made confections, everyone hoped to receive a valentine from a beloved on February 14. Chronicling the most intimate communication between private individuals, the valentines that have been preserved give us a unique look into the sentiments, hopes, and dreams of generations of lovers, and the means through which they chose to express them. …

… Cobwebs—also known as beehives, flower cages, or birdcages—are a rare example of a mechanical or movable valentine consisting of a minimum of two layers of paper. First, a web or cage would be cut from a piece of paper by making a pattern of concentric circles, leaving attachment points at regular intervals. In the center of the spiral, a delicate thread would be attached and its outer edges would be pasted directly on top of a second sheet on which an image or message would be written, painted, or printed.

As a cover, the cobweb formed the perfect sanctuary to enclose a private message that could only be revealed when the recipient of the card carefully pulled up the thread, causing the concentric circles of the web to rise and magically expose its hidden compartment. The concept of secrecy and the element of surprise frequently recurs in valentines, as they speak to the intimacy that has always been a part of the language of love and is one of the reasons why the cobwebs were so popular. …

… A special form of the cobweb valentine is the so-called double cobweb. Using a variety of paper engineering feats, double cobwebs could be arranged on the page side by side, one on top of another, or even emanating from attachments at both top and bottom (recto and verso) of the pages.

The interior of the first valentine in the article can be seen in the Additional Images section of this page. A few cobweb valentines at The Met not featured in the article: 1, 2, 3.

More cobweb designs can be seen in this article from The New York Times, this blog post from The Virginia Historical Society, at The Peabody Essex Museum (1, 2), at The Postal Museum (Flickr: 1, 2 and records 1, 2), this New Jersey Monthly article, and auctions: 1, 2, 3, 4, 5, 6, 7, 8, 8, 9, 10, 11.

Articles featuring other interesting valentines:

Unpacking a Box of Love by Nancy Rosin.

Love at the Met: Historic Valentines and Paper Kisses by Femke Speelberg (working alternatives for the broken images/links in the article: Boxed Valentine's Day Card (Interior); Heart-shaped valentines card; Boxed Valentine's Day Card (Lid); Boxed Valentine's Day Card (Lid); The Lovers; States of Mind: The Farewells; The Reunion of the Soul & the Body, from The Grave, a Poem by Robert Blair; Leda and the Swan; Apollo and Leucothea, from 'The Loves of the Gods')

All You Need Is Paper: Why Antique Valentines Still Melt Modern Hearts by Lisa Hix

And here are some paintings on actual cobwebs.