There is some interesting history on this:
ARM & HAMMER BAKING SODA The muscular arm and raised hammer represent Vulcan, Roman mythological god of fire and metalworking, and was once the symbol of the Vulcan Spice Mill in Brooklyn, N.Y. James A. Church, owner of the firm, took the symbol with him when he shut down the mill in 1867 and went into business with his father, a baking soda manufacturer. Shortly afterwards, the “power” behind Vulcan’s arm and hammer became the trademark of Church’s baking soda, which had the “force” to make bread dough rise.
We can only hypothesize why Church, Patterson, or anyone else decided to use the arm & hammer emblem as a commercial logo at this time. First, it was an established icon that would be easily remembered. As a symbol of the national labor movement, the arm & hammer still retained the positive connotations of integrity and strength derived from the Vulcan myth. This is also the exact time that the seeds of the union label movement were being planted across the country in San Francisco, starting with the Carpenter’s 8 Hour lumber stamp (1869) and the Cigar Maker’s “white men” label (1874). These labels were meant give consumers an economic way to support labor, and to a way to differentiate higher quality union made goods from those produced in sweatshops and other unsanitary conditions. It may be possible that consumers were prepared for the concept of the union label because they had already started to see and trust products that used “honest labor,” the arm & hammer emblem or other variations on this theme as a selling point. - http://www.likelihoodofconfusion.com/nailing-the-arm-and-hammer-trademark/
Church & Dwight Co, Inc. is a major American manufacturer of household products that is based in Ewing, New Jersey. While it manufactures many items, it is best known for its Arm & Hammer line which includes baking soda and a variety of products made with it. Church & Dwight was ranked #652 in the Fortune 500 listing of companies in 2017.[3]
The Arm & Hammer logo, which dates back to the 1860s,[5] is often incorrectly claimed to have originated with tycoon Armand Hammer. Hammer was so often asked about the Church & Dwight brand, however, that he attempted to buy the company. While unsuccessful, Hammer's Occidental Petroleum in 1986 acquired enough stock for him to join the Church & Dwight board of directors.[6] - https://en.wikipedia.org/wiki/Church_%26_Dwight
Armand Hammer...In his 1983 book, Red Carpet, author Joseph Finder discusses Hammer's "extensive involvement with Russia."[25] In Dossier: The Secret History of Armand Hammer, Edward Jay Epstein called Hammer "a virtual spy" for the Soviet Union.[26]
Hammer was a supporter of the Republican Party. He boosted Richard Nixon's presidential campaign with $54,000 in campaign contributions. He pleaded guilty to charges that one of these donations had been made illegally and received probation and a $3,000 fine,[29] but was later pardoned by Republican U.S. President George H. W. Bush
Hammer was very fond of Albert Gore, Jr., and in 1984 under Hammer's guidance Gore, Jr. sought Tennessee's Senate office previously held by Howard Baker. Hammer supposedly promised Gore, Sr. that he could make his son the president of the United States. It was under Hammer's encouragement and support that Gore Jr. sought the Democratic Party presidential nomination in 1988.[45][46]
In the 1980s Hammer owned a considerable amount of stock in Church & Dwight, the company that manufactures Arm & Hammer products; he also served on its board of directors. However, the Arm & Hammer company's brand name did not originate with Armand Hammer. It was in use some 31 years before Hammer was born.[47] He was spurred to buy shares in the company as a result of often being asked about the brand being so close to his name. - https://en.wikipedia.org/wiki/Armand_Hammer