Both of Mr. Dorsey's children, as he is known to refer to the companies, bear the imprint of their creator and his careful, trimmed-down aesthetic.
But Mr. Dorsey's eye for simple solutions has not been enough to turn around Twitter since he returned in 2015 to lead the company he helped found in 2006. He has failed to rein in the use and abuse of the service by trolls and political actors, and he has not found a convincing way to make money despite its ardent user base.
At Square, Mr. Dorsey has led one of Silicon Valley's more understated success stories, built from the little plastic white square that allowed small businesses to accept credit card payments via their iPhones. Square has grown into a much broader financial services company, despite some hiccups, like a much-heralded partnership with Starbucks that flamed out.
Although Twitter's overall revenue shrank 5 percent in the second quarter from the year before, expanding the company's losses, at Square, revenue grew 26 percent, moving the company closer to profitability.
The divergent fate of Mr. Dorsey's two companies can be understood simply as a matter of timing and control. When he began his second stint as Twitter's chief executive in 2015, the company was already struggling and had a series of internal problems that were hard to change, including executive turnover and competing strategic visions.
At Square, on the other hand, Mr. Dorsey was able to build the whole company from the ground up, with knowledge from Twitter of how things can go wrong.
"Square wasn't Jack's first rodeo," said Randy Reddig, who was on the founding team at Square in 2009. "It was obvious early on at the company that he took lessons learned from his time at Twitter and applied them at Square."
Mr. Reddig said that Mr. Dorsey's education from Twitter's fractious early years was particularly apparent in the control that Mr. Dorsey kept over hiring and compensation decisions at Square, with a focus on creating a loyal team.
Through Twitter and Square, Mr. Dorsey declined to comment for this story.
Mr. Dorsey's success at Square and his trouble at Twitter also tell a broader story about Silicon Valley, where the boring, back-end businesses often end up generating more money than the flashy social networks and consumer services that capture the public's eye.
Mr. Dorsey trained Square's sights on a mostly invisible business — electronic payments — that has momentum as more and more commerce moves online.
Capgemini recently estimated that electronic payments should grow 10.9 percent a year between 2015 and 2020. It is no coincidence that the most successful American financial start-up of the last decade, aside from Square, is Stripe, which helps online businesses take payments.
Beyond just payments, though, Square has taken aim at the much larger goal of providing a tech-savvy alternative to the big banks, expanding out from payments to lending and online deposits.
This summer, Square applied for a bank charter in Utah, one of just three so-called fintech firms to take such a bold step (the others are Social Finance and Varo).
Square is most visibly positioned to use a bank license with its business customers, but the company has also been building more banklike services for consumers, most notably with its app Square Cash.
Square Cash is often described as a competitor and imitator of Venmo, the popular app owned by PayPal that makes it possible for friends to send each other payments by smartphone.
In recent months, though, Square Cash has quietly bypassed Venmo to become the most frequently downloaded financial app of any kind on both Apple and Android phones, according to Apptopia. At various points over the past week, Square Cash was ahead of Venmo and Twitter on the charts of the most downloaded iPhone apps.
The big banks had aimed to challenge Venmo with their own service, Zelle, but that has not posed a credible challenge to either Venmo or Square Cash so far, the rankings show.
Mr. Dorsey and his team made several early design choices with Square Cash that helped make it different from Venmo. As would be expected from Mr. Dorsey, Square's app has a clean, green interface, compared to Venmo's busy blue dashboard.
Square also made an apparently boring technical decision, to use the debit card networks rather than bank transfers, to move money around. That has made it easier for Square Cash to put money instantly into the bank accounts of its users, and to collect a fee for the service.
These differences appear to have made Square Cash more popular with lower-income customers who more often need instant access to their money and who don't have as wide a variety of credit cards and other financial options.
"We are reaching an audience that may not have a bank account or may not have a full suite of services from a bank," Mr. Dorsey told analysts last quarter.
The shift in people using Square Cash as something more like a bank account has been an important one for Square's bottom line. While Square and PayPal generally don't make money when customers send money to one another, they can take a cut when customers start using the service to make payments at stores.
This all looks like a similar playbook to the one Square used with its original credit card dongle, starting with a simple service for an underserved audience and then building from there.
Keith Rabois, who was the chief operating officer at Square until 2013, said that the Cash app reflects Mr. Dorsey's ability to make deceptively simple products that answer a complicated need.
Even though the revenue from Cash is small compared with the money that Square pulls in from businesses, Mr. Rabois said that "it may be the most important thing Square is working on."
During Mr. Rabois's tenure at Square, the future did not always look so bright. The company's first attempt at a consumer application, Square Wallet, never took off and was discontinued, bringing down the Starbucks partnership with it.
Many analysts also expected that Square would have to reduce the price it charged businesses to process credit card transactions as it began working with larger businesses that had more options.
Over time, though, Square has been able to keep its prices stable by offering its business customers other services, such as payroll and scheduling.
A KeyBanc analyst recently polled 20 businesses that use Square and found that most of them had been offered lower card transaction fees by other companies but passed them up because of the add-ons that Square offers.
One of the most profitable such services is Square Capital, which offers small loans to businesses. The revenue from these loans has been increasing faster than the overall growth of the company and has Square looking more like a bank every day.
These days, one of the biggest concerns that analysts voice about Square is the time that Mr. Dorsey has to spend on Twitter, with all its problems.
During a talk at Silicon Valley's Computer History Museum in August, Mr. Dorsey said he thought he had proved that he didn't need to make a choice between the two.
"They both have different needs and are in different phases," he said. "Focusing on one thing doesn't mean you completely lose sight of something else."