What is the thesis of the church towards asexuality? What is the position of the Church on asexuals?

An asexual is a person who does not experience sexual attraction towards any person, thus does not desire to have sex. 

Asexuality is distinct from abstention from sexual activity and from celibacy,  which are behavioral and generally motivated by factors such as an individual's personal or religious beliefs.

An answer from the Church tradition is welcome. 

Does the Bible mention anything about asexuality? The church? Jesus himself? Is asexuality a sin and is it considered the same as homosexuality?
Plus one for a decent, albeit unusual, question.  

After reading your question for the first time, I thought immediately of Jesus' words to his disciples in the context of His teaching on divorce:


  "'For there are eunuchs who were born that way from their mother's womb; and there are eunuchs who were made eunuchs by men; and there are also eunuchs who made themselves eunuchs for the sake of the kingdom of heaven.  He who is able to accept this, let him accept it'" (Matthew 19:12 NASB, Updated Edition).


Compare Jesus' words with Paul's words in 1 Corinthians 7:1-9, where the apostle says, in effect, it is better to be single than to marry, but 


  "each man has his own gift from God, one in this manner, and another in that" (7:7b).  


Combine these two teachings with other teachings throughout the Scripture, which for brevity's sake I'll not cite, and the message is clear:  


Marriage is good and right and proper, and it is better to marry than to burn with lust and be tempted into sin by Satan because of a lack of self-control.
Singleness is good and right and proper, whether a person never marries or a person marries and is divorced, but then decides not to re-marry.  Each person has his or her gift from God, so there is no hard and fast rule as to whether to marry or not to marry. 
Neither marriage nor singleness is superior.  As Constable has noted, ". . . neither Jesus nor the apostles viewed celibacy as an intrinsically holier state than marriage (1 Tim. 4:1-3; Heb. 13:4; cf. 1 Cor. 9:5). They viewed it as a special calling that God has given some of His servants so they can be more useful in His service."
A eunuch can be a eunuch in the flesh (i.e., he is castrated), but he can also be a eunuch in spirit, and for the sake of the kingdom decides to remain single.  If he is so gifted, that is the sensible thing to do.  Married life has many distractions and complications that the single life does not, whether you are a single man or or a single woman!  A woman, too, needs to be guided by her "gift" in this regard, and there is nothing inherently bad about choosing to marry or inherently good about choosing not to marry.   


The number of Christians who are truly asexual is probably very small indeed.  The number of Christians who have a normal sex drive--neither weak nor strong-- but decide in effect to put their sexuality on hold for the sake of the kingdom is probably a little higher.  This is not to say they will never be tempted.  Moreover, in light of the Bible's silence on  masturbation, I am of the opinion that occasional, private, individual masturbation is understandable for these gifted believers and does not mean they should necessarily reconsider their calling. 

The number of Christians is likely very large, if it does not in fact constitute the vast majority of adult Christians worldwide, who have a normal sex drive (albeit of individually different strengths) and who neither desire nor have the gift to be single.  

In conclusion, God has a wonderful plan for all His children's lives, whether that plan involves marriage, singleness, or even the pain associated with divorce, remarriage, or widowhood.  The key is for us to function within the parameters of our giftedness, to do so in the strength and joy of the Lord, and to be content in all circumstances (Philippians 4:11-13), because as Paul said,


  "I can do all things through Him who strengthens me" (v.13).