According to the LXX, Leviticus 19:27 states:
You will not make curled hair from the hair of your head, nor shall you disfigure the appearance of your beard.
The Greek word for curl is σισόην which means to sift and is also used in Luke 22:31 in the context of Satan trying to sift Apostle Peter like wheat.
The surrounding context of 19:27 seems to be non-existent as this is the chapter that concerns general laws.
Does this mean that curling my hair is a sin? If so why?
First, the immediately surrounding verses of the text in question are significant. From verse 26 to 29, all those laws pertain to avoiding pagan beliefs and practices. Verse 27 is one of those.
Context helps sort out what was meant, as does going by the Hebrew word used in verse 27. From Matthew Henry's Commentary, he shows how (vs. 26) the way the Gentiles used blood was to gather the blood of their pagan sacrifices in a vessel and to identify themselves with demons. Superstitions and horoscopes etc had been seen by the Israelites when in Egypt; they must have no truck with them. Here is his quote regarding verse 27:
"There was a superstition even in trimming themselves used by the heathen, which must not be imitated by the people of God - You shall not round the corners of your heads. Those that worshipped the hosts of heaven, in honour of them, cut their hair so as that their heads might resemble the celestial globe; but as the custom was foolish itself, so, being done with respect to their false gods, it was idolatrous." Commentary, Matthew Henry, p 137 columns 2 & 3, Hendrickson, 2008
He then deals with other pagan customs, such as for the dead - cutting one's flesh, and marking one's body, also with prostituting one's daughter in pagan rituals. All these practices were of the one cloth - and had to be cut completely out of the people of God. So verse 17 is to be understood in that context - pagan customs.
Another commentary makes this observation about the verse in question:
"27 not round the corners of your heads: i.e. to cut round, so as to have a tuft of hair, like the Canaanitish priests. Cp. Jer. 9:26; 25:23; 49:32. corners of the beard = whiskers, as Egyptians did. Cp. Gen. 41:14." The Companion Bible (study notes) p161, Bullinger Publication Trust, 1974 reprint
There was no overt pagan association with making straight hair curly, or coiling it into ringlets any more than there was of making curly hair straight. The Hebrew word that has been taken in the LXX to mean 'curl' is naqaph in Young's Concordance, and means 'To go or compass round about'. That might lend itself to the idea of straight hair becoming 'round' in the sense of curly. But in paganism of the day, the emphasis was on cutting off hair in certain ways.
For Christians today, modesty in appearance and apparel is encouraged, but there is simply nothing said about making straight hair curly (or vice versa).
Actually, Lev 19:27 says nothing about curling the hair. Here is the text from the LXX:
οὐ ποιήσετε σισόην ἐκ τῆς κόμης τῆς κεφαλῆς ὑμῶν, οὐδὲ φθερεῖτε τὴν ὄψιν τοῦ πώγωνος ὑμῶν. = Ye shall not make a round cutting of the hair of your head, nor disfigure your beard. (Brenton)
Ellicott comments on this verse as follows:
(27) Round the corners of your heads.—That is, they are not to shave off the hair around the temples and behind the ears, so as to leave the head bald except a dish-like tuft upon the crown, thus imparting to their heads the form of a hemisphere. This was done by the Arabs, and other worshippers of the god Orotal. Hence the Arabs are ironically called “those with the corner of their hair polled,” as it is rightly rendered in the Margin (Jeremiah 9:26; Jeremiah 25:23; Jeremiah 49:32).
Barnes is similar:
Round the corners of your heads - This may allude to such a custom as that of the Arabs described by Herodotus. They used to show honor to their deity Orotal by cutting the hair away from the temples in a circular form. Compare the margin reference.
The clue is in the Jeremiah references ,especially ch9 v26, where those who cut their hair in odd ways are asociated with the uncircumcised. It is about using the wrong kind of identification marks. Circumcision helps to identify a member of God's people Israel. Cutting the hair was at the time the mark of certain tribes who did not worship YHWH. This is one of those laws that are related to the immediate social context. The act is not a sin to those who "serve not under the old written code but in the new life of the Spirit" (Romans ch7 v6, RSV).
Lessons from Archaeology Archaeology has come a long way in the past hundred years. We have, not only relics, but monuments and murals, along with statues, from which we can draw pictures if the ancient people groups...and Empire rulers.
A scan of these pictures in Archaeology books shows the cosmetic features of the ancient peoples that the Israelite nation intermingled with. Kings with "ringlets" or some with wavy hair were common. Beards manicured with stylish features are seen throughout. Assyrians, Babylonians, Egyptians, Canaanites all engaged in barber shop manicuring.
But throughout the Patriarch and Mosaic eras God warned the Israelites to maintain their unique different identity. They were not to become like the peoples they were to encounter. Neither adopt their gods, unclean eating habits, immorality...or cosmetic features and styles. The early Israelites were agrarian, and manicuring their hair and beards was nor first on their list in everyday work. Grooming was simple. Full beards, and hair bunches...not too long and not too short were all that was necessary. They weren't styled to attend palace banquets...just attend sheep and goats! This identity was to remain theirs. And it would help keep them separate and prevent the culture drift that clans often succumb to.
Modern Application While modern Christians are not obliged to keep the details of the Law--it has been nailed to the Cross--yet the principles found there are to be observed: Piety, morality, integrity, charity, family, etc. These passages (about hair and beards) highlight the tendency of young people to want to conform to the culture around them. Preventing assimilation to surrounding culture is a constant struggle between parents and their children. Moses's decrees give strength to parental authority. Culture drift is a constant danger! And any help is to be appreciated.
While the Torah dealt with "men" and their attire, the New Testament Apostles Paul and Peter dealt with "women" and their cosmetics habits* as well:
In like manner also, that women adorn themselves in modest apparel, with shamefacedness and sobriety; not with broided hair, or gold, or pearls, or costly array; but (which becomes women professing godliness) with good works. 1 Timothy 2:9-10)
Likewise, you wives...whose adorning let it not be that outward adorning of plaiting the hair, and of wearing of gold, or the putting on of apparel. But let it be the hidden man of the heart in that which is not corruptible, even the ornament of a meek and quiet spirit, which is in the sight of God of great price. (1 Peter 3:3-4)
Both with men and women, there is to be no "legalistic approach" nor judging of one another over style---different cultures differ---but the awesome, God-inspired principles of modesty, simplicity, holiness, moderation, and ministry, etc. are to remain in the forefront of conduct.
Parents (as well as pastors) are obligated to be thankful to Moses for bringing this to our awareness in his Pentateuch. We are a holy people, a prized possession, a godly kingdom...
Addendum
Linguistic Research It should be noted that the verse quoted by the OP was an English translation of a Greek translation of a Hebrew text! The LXX translated by Alexandrian Jews does not always follow the (what we now call) the Masoretic Hebrew wording. Notice the modern translations into English from that Hebrew text:
Do not cut the hair at the sides of your head or clip off the edges of your beard. (NIV)
Ye shall not "round" the corners of your heads neither shalt thou mar the corners of thy beard. (KJV) ["Round" is an old English word meaning "to trim hair short around the head" THOMPSON CHAIN REFERENCE BIBLE, p. 1808]
