PORT ST. LUCIE — The Mets and Yankees exist together and apart. They share a city, yet are hardly ever there at the same time. They train in the same state, but only began playing each other again in spring 2012 after a 16-year divorce.

They played the East Coast end of their Florida home-and-home Wednesday, and getting together in the same place at the same time actually made me wonder about what they have in common. Let us count some ways:

So far both teams have liked what they have seen from Neil Walker and Starlin Castro. Walker, though, is probably a one-and-done for the Mets. Castro is signed through at least 2020, and the Yankees are hoping he joins Didi Gregorius in a long-term middle-infield solution.

David Wright has yet to play a game this spring as the Mets express caution about trying not to push too hard too soon due to his spinal stenosis. He is not expected to get into a game before next week, and there has to be some questions about his readiness for the season — especially for cool weather in April — and then how much he will play overall.

At the moment, Wilmer Flores is the Mets’ backup at third. But in an ideal world, the Mets would have a lefty bat to allow Wright’s more frequent rest days to be against tough righties.

Chase Headley led the Yankees in games played last year despite not playing well on either side of the plate. His back condition is not as bad as Wright’s, but it is a factor. And the Yankees do not have a clear backup. They have liked what they have seen during the initial transition of Rob Refsnyder to third.

But the Mets and Yankees both have scouts eyeing third-base depth options. This is just my guesswork, but guys who can hit lefty such as Ryan Flaherty, Derek Dietrich, Yangervis Solarte and Andres Blanco could make sense for the Mets, and those plus, say, Cody Asche and Hernan Perez could be enticing to the Yankees.

Ultimately, the Mets and Yankees just decided the asking price was too favorable to ignore Yoenis Cespedes and Aroldis Chapman.

So after being concerned about Cespedes as a full-time center fielder, the Mets ultimately re-signed the righty slugger when he was amenable to doing a three-year contract with a one-year opt out.

It took weeks for Hal Steinbrenner to approve the trade for Chapman even after the Reds said they were agreeable to an underwhelming return package. The Yankees owner took his time to decide if a player who was alleged to be involved in a domestic abuse altercation (but never charged) and who almost certainly was facing a suspension was worth obtaining. Ultimately, the Yankees decided they were willing to hold their nose and accept the talent.

The Yankees might have one of the best endgames ever with Chapman, Dellin Betances (who is throwing so much better this spring than last) and Andrew Miller. But they already know Chapman is going to miss the first month of the season. Chasen Shreve, after a terrible finish to last season, has thrown well. But the Yankees had many relievers circulate in and out in 2015 without success, and many are back this year. Nick Rumbelow might be the favorite among that group. Kirby Yates has caught the Yankees’ eye.

But an interesting possibility is someone who ties the Mets to the Yankees: Luis Cessa was a piece the Mets included to Detroit in obtaining Cespedes last July. The Yankees obtained him with Chad Green at the Winter Meetings for Justin Wilson. The plan was to put Cessa at Triple-A for rotation depth. However, his easy 95 mph fastball, sinker and poise have caught the attention of both scouts and Yankees brass, who will consider him for the bullpen.

The Mets want to believe a group of Addison Reed, Hansel Robles, Antonio Bastardo and Jerry Blevins can successfully get the ball from their terrific rotation to Jeurys Familia. Keep an eye on Josh Smoker. The Mets love his stuff, though they might make him go to Triple-A to just add one more level of confirmation before bringing him to the majors.

Both New York teams believe they have a legitimate prospect ready to back up their starter. The question is whether they see enough at-bats for Kevin Plawecki and Gary Sanchez to keep them sharp and productive, or if they are better off letting both play regularly at Triple-A.

The Mets are mulling lefty Johnny Monell as Travis d’Arnaud’s caddie. The Yankees are considering Carlos Corporan and Austin Romine to back up Brian McCann, and Romine early on has stepped ahead in that competition. But the Yankees want to deliver more consistent threat against lefty pitching. They think Castro, Aaron Hicks and a healthy Mark Teixeira help, but that Sanchez could also be an asset in that area.