Let's learn about "tsun-dere(ツンデレ)".
http://image.news.livedoor.com/newsimage/f/2/f2b6a_889_f16debb5845a015...https://en.wikipedia.org/wiki/TsundereSo, that is same meaning but using "not straight(often quite oppsite)" meaning words/phrases.
It is one of "emotional expression" which is often seen in teen or young age.
That "あげる(ageru)" is expression of the meaning "I am superior (upper position)",
= "I will give it(invitation) to you, if you want/wish/hope it or say "Yes!".
Other phrases,
(V)して(orさせて)あげても-よい(-構:かまわない)
(V)して(orさせて)も-よい(-構わない)
"Lady (お嬢様 daughter of celeb family)," phrases,
(V)して(orさせて)あげて(も)-構まわなく(っ)てよ!
(V)して(orさせて)あげて(も)-よく(っ)てよ! using a lot. (V = Verb)
Because the expression of affection of the Japanese is very delicate,
so is the character of Tsundere (mainly girls),
but also a lot of "insensitive(鈍感:don-kan)" boy/guys
written into the story of "love comedy(ラブコメ:love kome)".
You can find it easiest into lines of Rin 2nd season of "Fate/stay night UBW"(ep16 is very famous),
https://goo.gl/pgXSLqalso "Hitagi Senjogahara" in "Mogatari" series("Tsundere serviiice!" Bakemonogatari ep14 is very famous),
https://en.wikipedia.org/wiki/Monogatari_(series)
"Houki Shinonono" in "IS(Ai-Esu)" are one of them and very famaus and popular.
https://en.wikipedia.org/wiki/Infinite_StratosUsing tsundere is not only in girl/young woman phrases,
but there are few cases that it is called tsundere, bc just not looks like "cute/lovely".
The essence of tsundere is "I like/I'm loving" that is overflowing from thorn words.
Tsundere characters are easy to become "thin books",
but if you are not an adult, will be scolded by "Miss. Mahoro". :)
https://en.wikipedia.org/wiki/MahoromaticTsundere is one of "Gap Moe(ギャップ萌え)".
https://hinative.com/ja/questions/158606A person who were contacts suddenly wearing glasses,
changing hairstyles and changing the evaluation is a common phenomenon in the world?
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And,
「私はあなたをパーテイーにさそってあげる. 」 is not perfect copy.
「私はあなたをパーティーにさそってあげる。」 is.
Did you find what is different?
However, no need "私は" if in talk, and "私が" is best.
("私は" is not wrong, but not fit as sounds of Japanese)
When you want to identify individuals, "が" using usual(not for all).
Sample of difference :
私はそれを食べた。 "I ate it."
私がそれを食べた。 "I ate it"
First has also including "I have eaten it before.(私はそれを食べたことがある。こと=経験 : experience)".
But second cannot translate like that.
Opposite sample :
私は今どこに居るのだろう? "Where am I, now?"
But it cannot use(to say and write),
私が今どこに居るのだろう?
However, second can use as "私が今居る所は、どこ(なの)だろう?". (I wondering "Where is this location?".)
Also,
私は電車を使います。 "I take a train." (as daily life)
私が電車に使います。 "I will take a train."
(in specific case as selection, someone take a another tranport, taxi, bus, air etc)
In this case, both "Subject" is specified something.