Twiss, the former Manchester United youth team player, was a class above and drove on a Morecambe side who, like Halifax, are struggling with the weight of expectation this term.
Morecambe's left-back Michael Howard was sent off in the 23rd minute for two bookable offences, but Halifax failed to take advantage.
Former FC United winger Steve Torpey, who impressed Halifax boss Chris Wilder so much in a pre-season friendly that he snapped him up, made his first start on the left.
He showed some good early touches and scored a superb opener when he curled home left-footed from the edge of the box on the stroke of half-time.
Halifax, though, could not build on their lead and Twiss ensured a hard-earned point with a free-kick special of his own.
Morecambe started at a high-tempo and Craig Stanley saw an early shot cleared off the line by Matt Doughty.
In the 13th minute referee Mo Matadar was lenient on Howard when the Shrimps defender appeared to push over Ryan Sugden in retaliation for a foul.
With the Halifax fans baying for a red card Mr Matadar produced yellow, but ten minutes later Howard did see red when he tugged back Danny Forrest who had been set to skip away down the right.
The shirt pull in itself was probably not enough to earn a booking and maybe Mr Matadar realised he should have sent Howard off the first time and ensured justice was done.
Despite the dismissal, Morecambe were the more attack-minded side and home goalkeeper Craig Mawson tipped over a piledriver from Danny Carlton.
Halifax pressed just before the break and Lewis Killeen twice wasted good opportunities before Torpey broke the deadlock.
A sweeping crossfield move saw Sugden allow the ball to reach Torpey, who cut inside and measured a finish which curled round Ryan Robinson's out-stretched arm and nestled into the far corner.
Even with a goal on the board Halifax found it tough and Morecambe continued to attack.
A well-flighted Twiss corner was perfect for Carlton but his header zipped just wide.
The second half was littered with free-kicks and stoppages and tempers were never far from the surface.
In the 61st minute Morecambe levelled when Twiss produced a free-kick from the top drawer.
The Halifax wall stood its ground but Twiss curled the ball round and into the far corner, with Mawson motionless.
Shaymen substitute Andy Campbell twice went close to securing all three points, but in the end Wilder's men had to settle for a draw.