“It’s Thursday the first time I see it,” says Edward, the narrator of “Lily and the Octopus.” “I know it’s Thursday because Thursday nights are the night my dog, Lily, and I set aside to talk about boys we think are cute.” So begins Steven Rowley’s startlingly imaginative and just as tender debut novel about a guy, his dachshund and the fight of their lives. “It” is the uninvited cephalopod of the title, and it sits atop Lily’s velvety head. Edward just can’t bring himself to say “tumor.”

“Lily and the Octopus” is a love story sure to assert its place in the canine lit pack. It’s a saga of grief, too. And Rowley reminds us with his wild conceit that all love stories are grief stories at some point.

Edward is 42 years old. His screenwriting career is stalled. He’s gay and lives in Los Angeles, where being 40 and stuck can be especially uncomfortable. He has a best friend, Trent, who knows when to order him a martini and offer a Valium chaser. He has a therapist he doesn’t much care for. He has an ex-boyfriend, Jeffrey. He has a mother whose concerned phone calls don’t assuage his sense of not being loved enough by her. He has 12-year-old Lily.

Edward recounts the origins of their love. “I remember Lily licking the tears from my face,” he says about bringing the puppy home and starting to cry. “THIS! EYE! RAIN! YOU! MAKE! IS! FANTASTIC! I! LOVE! THE! SALTY! TASTE! YOU! SHOULD! MAKE! THIS! EVERY! DAY!” Yes, she’s fond of the exclamation point — and how!

The octopus speaks for the first time shortly after Lily’s seizures begin. “If you’re talking to her, she’s asleep,” he tells Edward. He’ll have plenty more to say, often with an infuriatingly droll delivery.

He’s a jerk, but he isn’t the only smart aleck here.

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Edward admits early on he’s prone to sarcasm. And Rowley handles the reliable-not-so-reliable tango of his narrator deftly. This trait comes out most often when he’s talking about (or even with) his therapist, Jenny. “I filter her analysis through the voice of my preferred, imaginary therapist and he comes up with sharper advice.”

The novel’s structure is marvelously idiosyncratic. There are lists: “Eight Times I Was Cowardly” and “A Complete List of Lily’s Nicknames.” Sections hew to their own logic, unfolding chronologically or thematically. The most metaphysically intricate section, “Ink,” touches on blindness and chew toys, tattoo artists and philosophy as well as offering a riff on the inventor of the Rorschach test.

As Lily grows sicker, Edward will engage the octopus in hand-to-tentacle combat. Sometimes his strategies are the stuff of lunacy: He visits a fishmonger in Chinatown and returns home with a dead octopus, a glint in his eye and a cleaver in hand. Other times, the stuff of literature. In “The Pelagic Zone,” where Edward and Lily board the boat “Fishful Thinking” and hunt for the octopus, there’s a chapter called “The Old Lady and the Sea.”

In recounting Lily’s life as he battles to save it, Edward captures 12 years of his own: He takes account of who he is, who he has become, who he can be. This self-knowledge comes by way of his darling Lily but also that damn octopus.

In telling their tale, Rowley reminds us just how strange it is that dogs have such abbreviated lives compared with their human beneficiaries. Oh Sneaky Creator of Meaning, why is this if not to school us about love and loss and the possibility (obligation?) to love again and lose again?

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Lily gives Edward so many lasting treats. Rowley has done the same. Be prepared for outright laughs and searing or silly moments of canine and human recognition. And grab a tissue: THERE! WILL! BE! EYE! RAIN!