I agree with YogoZuno above, the rules are not very specific on this--and I think it's to allow for a variety of playstyles and flavors. We don't all want to be a robed guy walking into battle holding a shining holy symbol above our heads shouting and summoning the will of a particular god in a booming voice and shattering skeletons helter-skelter. There are different gods as well, sure Sarenrae might appreciate this, but what about Norgorber?
The cleric section mentions holy symbols only twice and it's only in the channel section. The section reads:
Regardless of alignment, any cleric can release a wave of energy by channeling the power of her faith through her holy (or unholy) symbol.
This first, to me, says nothing of needing to show the symbol at all, just that the cleric needs to have it in order to channel.
The second reference is:
A cleric must be able to present her holy symbol to use this ability.
Forums abound on debates about what "present" here means. Aside that this line seemingly, does not take into account those clerics that worship ideals and not a specific diety, it presents several questions. Does "present" mean the use of a free hand? Does it mean visible or not? Again, I would say it's unspecific on purpose. But, put together with how it would generally be read and how the class works, I think it is safe to interpret here that (again) generally the symbol would be visible--but not necessarily so.
Whether or not would be GM discretion and also it would be diety dependent, i.e. Norgorber might like a deceitful use of the holy symbol, while Sarenrae might not. That said, if the character were on a holy mission (or a mission for the greater good/evil) perhaps the diety could forgive the use of deception.
As noted above the cleric entry does not mention using a holy symbol to prepare spells, and so it isn't required. There are spells that require a "divine focus," which is explained as follows:
A divine focus component is an item of spiritual significance. The divine focus for a cleric or a paladin is a holy symbol appropriate to the character’s faith.
It goes on to state that:
The divine focus for a druid or a ranger is a sprig of holly, or some other sacred plant.
Again, clerics that worship ideals are left in the grey zone. This GM would rule that they have some sort of focus as described in the ranger/druid section.
However, once again, the holy symbol must be present to cast the spell. Whether it needs to be in hand and/or visible is, once again, GM discretion.
Another note is that the Channel ability is based on the cleric's Charisma score, both the save and the amount of times per day they can use the ability. As per the Cleric entry in the CRB:
A cleric may channel energy a number of times per day equal to 3 + her Charisma modifier.
Perhaps it is truly the charisma of the character, the physical enaction of their belief (in ideal or diety) that is enough of a display--so long as they have their holy symbol on them--it is they that unleash the divine energy.