My short answer to your question is . . . I don't know.

Now for some educated guesses which are true to the Scriptures.  Jesus told a parable about two men who went to the temple to pray.  (See Luke 18:10-14.)  The one was a self-righteous Pharisee who prayed about himself, citing all his religious good deeds and comparing himself favorably with others whom he looked down upon as "sinners."  (See verses 11, 12.)  If he looked down on others whom he considered below him, he probably looked up very confidently to God when he prayed.  Jesus said that his prayer did not result in justification before God because he exalted himself rather than humbling himself before God.

In the same parable, a Publican, i.e. a despised and corrupt tax collector, also went to the temple to pray.  He knew his unworthiness to approach God, and so he stood at a distance.  He beat his breast as a gesture indicating he recognized all he deserved from God was punishment.  He would not even lift his eyes in prayer, which must mean that his head was bowed in humility and shame.  His prayer was one for mercy.  (See verse 13.)  Jesus said that this man went home justified.  Jesus closed this parable with the following statement:  "For everyone who exalts himself will be humbled, and he who humbles himself will be exalted."  (vs. 14)

I don't know the significance of putting hands together in prayer.  I would speculate that it is a posture symbolizing focus.  As for bowing heads in prayer, I believe this is a posture of humility.  We must always come before God in humility as "God opposes the proud but gives grace to the humble."  (1 Peter 5:5)

Having responded to your question as best I can, I would go further in saying that I do not believe the posture of putting hands together and bowing the head are necessary for prayer.  In fact, I almost never see anyone assume this posture in prayer, and only rarely (and unconsciously) have I caught myself doing it.  Humility in prayer is a posture of the heart and not a physical posture.  Furthermore, the parable of the Pharisee and the Publican was really about the humility required of one who is looking to God for mercy and justification, or salvation.  For those who have received grace, mercy, and salvation through faith in the redeeming work of Jesus on our behalf, humility in prayer assumes another picture altogether. While we don't come to God boasting of our own goodness, we do come celebrating what Jesus has done for us.  He has reconciled us to God, given us new birth into the family of God, made us partakers of God's divine nature, and justified us, i.e. made us righteous as though we had never sinned.  The result?  We have a new approach in prayer:  "Let us then approach the throne of grace with confidence, so that we may receive mercy and find grace to help us in our time of need."  (Hebrews 4:16) 

As the redeemed people of God, humility partners with celebration.  Humility is to acknowledge that we have no righteousness of our own that would fit us to approach a holy God, but that same humility embraces the forgiveness and transforming grace that declares us righteous through Jesus' substitutionary death for us.  In short, the believer in Jesus can come to God in humility while coming with a head that is raised, eyes that are opened, and a heart that is jubilant. 

I trust that this response is a help to you and an encouragement in your own prayer life.  For the believer, prayer is a blessed opportunity to commune with the Almighty who is our Father in heaven and who desires to enrich our lives with everything necessary to sustain us in life and in godliness.  So, I trust that you will find yourself approaching God's throne in prayer with boldness, confidence and joy.