Most classic-rock fans would agree that the hard-charging “American Woman” is the most recognizable and lasting song from Canadian rockers The Guess Who. Not too shabby, considering that they also charted with a slew of other FM radio standards in the late ‘60s and early ‘70s. The anti-U.S. protest lyrics of “Woman” didn’t even stop it from reaching No. 1 on the charts here in early 1970; Lenny Kravitz's remake made it a massive hit again nearly 30 years later.

But lest one think the song came fully formed into the studio when it was recorded by Burton Cummings (vocals), Randy Bachman (guitar/vocals), Jim Kale (bass) and Garry Peterson (drums), think again. According to Peterson, the epochal tune simply started out as an impromptu jam.

“We were playing two shows in one night, and there was a break between them,” he recalls. “We couldn’t find Burton to start the [next show], so I started playing a groove, a Buddy Miles kind of riff on the double bass drum. I was really into the Electric Flag album at the time.

“Then Randy started playing that riff, and then Jim fell in, and Burton just came running on the stage and sang whatever was in his head,” continues Peterson. “We had just come back from touring the U.S. and saw all the unrest about race and the war in Vietnam. I mean, we were just happy to be back in Scarborough, Ontario, Canada playing a show in a curling rink!”

Peterson — who will admit that Bachman and Cummings have slightly different versions of the song’s origins — says that audience reaction was great, and the band continued to refine the lyrical and musical content over subsequent live shows.

“‘I don't need your war machines/ I don’t need your ghetto scenes’: It’s personifying the country as a woman,” he laughs. “And when you first sail into New York harbor, what do you see? The Statue of Liberty!”

A footnote comes in that “American Woman” and its flip side, “No Sugar Tonight/New Mother Nature,” became a double-sided No. 1 record. Other acts to accomplish that rare feat include Elvis Presley, the Beatles and the Rolling Stones.

And in a strange postscript, in July 1970, the Guess Who was invited to perform at the White House for Richard Nixon and his family. Needless to say, someone in the government asked them to drop "American Woman" from the set list, which they did.

Today, Peterson leads the 2015 edition of the Guess Who alongside fellow original member Jim Kale, as well as with Derek Sharp (vocals/guitar), Will Evankovich (guitar), and Leonard Shaw (keyboards). They’ll be playing a slew of the band’s other hits like “These Eyes,” “Laughing,” “Undun,” “Hand Me Down World,” “Share the Land,” and their ode to a certain late gravelly-voiced DJ, “Clap for the Wolfman.”

The Guess Who’s origins stretch back even further than their 1966 formation. Peterson, Kale, and Bachman, all from Winnipeg, were part of a group called Chad Allen and the Reflections, who had scored a Canadian hit with a cover of Johnny Kidd and the Pirates’ “Shakin’ All Over.”

However, rather than put the band’s name on the single, the record company — in an attempt to build some PR mystique – credited it to “Guess Who?” Maybe foolhardy kids would think it was a British Invasion group masking their identity, the thinking went. Or even the Beatles themselves.

Keyboardist Bob Ashley left after the single was released and replaced by Cummings. When Allan himself left the next year to continue his higher education, The Guess Who (without the question mark now) began their career.

Except for sporadic previous reunions, Bachman and Cummings are long gone, and the band has weathered many lineup changes and periods of hiatus since their formation. Today Peterson and Kale keep it going as not only the living links to the Guess Who’s heyday, but as a tight rhythm section all their own.

“It is a comfort to look over and see him there,” Peterson laughs. “I mean, we’ve been playing together on and off for 53 years! He’s 72 and I’m 70!”

And while Bachman and Cummings received the written songwriter’s credits on the bulk of the band’s hits, Peterson feels shortchanged – both literally and figuratively – when comes to royalties over the years. It’s certainly not an uncommon complaint among band members of many classic-rock groups, few of whom could imagine that a song could be commercially popular and vital decades after its release, making a steady source of income long after a band's recording and touring peak.

“The four of us created the music,” he reflects. “I mean, Burton and Randy, they wrote the music, but had they done those songs like they [originally] wrote them, they would not have been hits. Not even close.”

Instead, Peterson likens a finished Guess Who song to the end product of a meat grinder, where multiple ingredients and contributions make the final tasty product that people know and love and keep coming back to consume.

“The songs would come so basic to rehearsal, and they would morph. Everybody contributed something,” he sums up, adding that perhaps credit should have been amended in subsequent years to include himself and Kale.

“It’s a fine line about the [registered] credits. When they pay out performance royalties when a song is played on the radio or is on TV or in a movie, only the [registered] songwriters get it.”

Peterson is happier to discuss Houston – fondly recalling gigs here in the ‘80s and ‘90s at Rockfeller’s (“that place that was a bar inside a bank”). But he says the Guess Who's current lineup has an even closer tie to the city. Houston is where Derek Sharp made his hastily-arranged debut as the band’s singer/guitarist after their previous front man was in a life-threatening accident in Australia.

He remembers the gig, a private corporate show at the Woodlands Pavilion in 2008, vividly.

“We sat in the room with him with an acoustic guitar right before the show, and that was all the rehearsal we had!” Peterson laughs. “I was terrified, but he was marvelous!”

The Guess Who performs at 5 and 8:30 p.m. Saturday, October 31 at Dosey Doe, 25911 I-45 N., The Woodlands.