Bill Mumy and Jan Handzlik took different routes in their careers, but they share a common bond: acting in classic episodes of “The Twilight Zone.”

Rod Serling’s black-and-white (mostly horror) anthology series, which originally aired on CBS (1959-64), will return Monday in the form of Jordan Peele’s “Twilight Zone” revival, streaming on CBS All Access with stars including Seth Rogen, Chris O’Dowd, Sanaa Lathan, Greg Kinnear and Tracy Morgan.

Mumy (as “Billy” Mumy) was in three “Twilight Zone” episodes — most memorably in 1961’s “It’s a Good Life” as the thought-reading demon seed Anthony Fremont. He also co-starred in “Long Distance Call” (1961) — as a 5-year-old talking to his dead grandmother over a toy telephone — and with Jack Klugman in 1963’s “In Praise of Pip.”

Handzlik, now an investigations and trial lawyer in LA, was 15 when he shot the 1961 mob-mentality episode “The Monsters Are Due on Maple Street.” He played Tommy, who stirs deadly neighborhood hysteria by suggesting outer-space aliens have caused the weirdly inexplicable events on Maple Street when the power goes out. (Turns out he was right.)

I spoke to Mumy, 64, and Handzlik, 73, about their “Twilight Zone” roles — and what it was like to appear in one of the most acclaimed series in TV history.

Mumy: I learned a very important lesson from Cloris Leachman on “It’s a Good Life.” One of the very first closeups we were shooting was of [Leachman] inside the house as I was delivering offstage dialogue to her. I was goofing around — I remember I was kind of making funny faces at her — and she said, “Cut!” and she took me aside. It wasn’t an embarrassing moment, but I’ll never forget it. She said, “Now listen, when you’re off-camera you have to be just as good as you are when you’re on-camera. You can’t just goof around.” Look, I was 6 or 7 and the bottom line is [that] nobody ever said that to me. I was like to myself, “Oh, OK, I get it. I see exactly what you mean. That will never happen again.” And it never did.

Handzlik: I have a very distinct memory of the cast members sitting around a table going over the script with Rod Serling. I can’t remember anything he said, but he personally participated in at least the dialogue rehearsals. I really enjoyed working with [co-stars] Jack Weston, Claude Akins and Barry Atwater. I remember Jack Weston was very funny and was always cracking jokes.

Mumy: Jack Klugman was so great, and so passionate, and we were filming out in Santa Monica at Pacific Ocean Park at night, which was really creepy even though it was my home turf and I used to go there all the time. We were about to film this scene where he grabs me and he can’t believe I’m physical and real and there and he starts hugging me and kissing me. Both of my parents had accompanied me that night and Jack … came up to them, which he did not need to do, and explained the passion of the scene. I remember him saying to my dad, “I’m really gonna grab your kid and really gonna lay one on him and I want you to be comfortable with it.” I’ll see that scene today and that’s where my mind goes.

Handzlik: It was a lot of fun by virtue of what happened in the episode with the lights going on and off and that’s where there was a little difficulty on the soundstage because we had to [shoot that scene] a number of times. Things didn’t start and stop the way they were supposed to.

Mumy: There’s a scene at the beginning of “It’s a Good Life” where Anthony has made a three-headed gopher and he’s holding it up and playing with it. It was so repellent to me — it was a legitimate prop; it was rubbery and had skin peeling off of it and was a disgusting prop. I remember shooting that [scene] and just grimacing, like “Oh God, do I have to hold this!”

(On Rod Serling) What I recall very clearly was that he was a very light presence on the set. You caught him in that black suit, but he didn’t feel like a black suit kind of guy … and then, of course, him filming those intros and outros. I was there for a couple of those. He stopped production [on “Long Distance Call”] for about 45 minutes and rewrote that last speech where my dad, played by Phil Abbott, picks up the toy telephone and pleads with his mother to not let the kid die. Rod went off to a corner — I can actually see him behind a set with 2-by-4s coming down behind the wall — and he rewrote that speech … and made it so much better.

I’ve done a lot of TV shows and certainly a lot of shows as a kid and they all have a fond spot in my memory. But the “Twilight Zones” hold my attention and they stay vibrant. I never cringe at them. I’m pretty happy to be a part of what’s inside that little frame.

“The Twilight Zone” premieres April 1 on CBS All Access with a double feature: a remake of “Nightmare at 30,000 Feet,” featuring Adam Scott (“Parks & Recreation”) in the role William Shatner created, followed by Kumail Nanjiani and Tracy Morgan in “The Comedian.”